-
UNIVERSITY Of I
-X CALIFORNIA I
A TREATISE
ON
ELECTRICITY AND MAGNETISM
MAXWELL
VOL. II.
Honfcon
MACMILLAN AND CO.
PUBLISHERS TO THE UNIVERSITY OF
Clareniron
A TREATISE
ON
ELECTRICITY AND MAGNETISM
BY
JAMES CLERK MAXWELL, M.A.
LLD. EDIN., F.R.SS. LONDON AND EDINBURGH
HONORARY FELLOW OP TRINITY COLLEGE,
AND PROFESSOR OF EXPERIMENTAL PHYSICS
IN THE UNIVERSITY OF CAMBRIDGE
VOL. II
AT THE CLARENDON PRESS
1873
[All rights reserved]
v.
.
J/VV*Wt
CONTENTS.
PART III.
MAGNETISM.
CHAPTER I.
ELEMENTARY THEOEY OF MAGNETISM.
Art. Page
371. Properties of a magnet when acted on by the earth .. .. 1
372. Definition of the axis of the magnet and of the direction of
magnetic force 1
373. Action of magnets on one another. Law of magnetic force .. 2
374. Definition of magnetic units and their dimensions 3
375. Nature of the evidence for the law of magnetic force .. .. 4
376. Magnetism as a mathematical quantity 4
377. The quantities of the opposite kinds of magnetism in a magnet
are always exactly equal .* .. .; .. 4
378. Effects of breaking a magnet .. .. 5
379. A magnet is built up of particles each of which is a magnet .. 5
380. Theory of magnetic 'matter' 5
381. Magnetization is of the nature of a vector 7
382. Meaning of the term 'Magnetic Polarization' 8
383. Properties of a magnetic particle 8
384. Definitions of Magnetic Moment, Intensity of Magnetization,
and Components of Magnetization .. .; .. .. .. 8
385. Potential of a magnetized element of volume 9
386. Potential of a magnet of finite size. Two expressions for this
potential, corresponding respectively to the theory of polari
zation, and to that of magnetic 'matter* 9
387. Investigation of the action of one magnetic particle on another 10
388. Particular cases 12
389. Potential energy of a magnet in any field of force 14
390. On the magnetic moment and axis of a magnet 15
812246
vi CONTENTS.
Art. Page
391. Expansion of the potential of a magnet in spherical harmonics 16
392. The centre of a magnet and the primary and secondary axes
through the centre 17
393. The north end of a magnet in this treatise is that which points
north, and the south end that which points south. Boreal
magnetism is that which is supposed to exist near the north
pole of the earth and the south end of a magnet. Austral
magnetism is that which belongs to the south pole of the earth
and the north end of a magnet. Austral magnetism is con
sidered positive 19
394. The direction of magnetic force is that in which austral mag
netism tends to move, that is, from south to nortb, and this
is the positive direction of magnetic lines of force. A magnet
is said to be magnetized from its south end towards its north
end.. 19
CHAPTER II.
MAGNETIC FORCE AND MAGNETIC INDUCTION.
395. Magnetic force defined with reference to the magnetic potential 21
396. Magnetic force in a cylindric cavity in a magnet uniformly
magnetized parallel to the axis of the cylinder 22
397. Application to any magnet 22
398. An elongated cylinder. — Magnetic force 23
399. A thin disk. — Magnetic induction 23
400. Relation between magnetic force, magnetic induction, and mag
netization 24
401. Line-integral of magnetic force, or magnetic potential .. .. 24
402. Surface-integral of magnetic induction 25
403. Solenoidal distribution of magnetic induction .. .. .. .. 26
404. Surfaces and tubes of magnetic induction 27
405. Vector-potential of magnetic induction 27
406. Relations between the scalar and the vector-potential .. .. 28
CHAPTER III.
PARTICULAR FORMS OF MAGNETS.
407. Definition of a magnetic solenoid 31
408. Definition of a complex solenoid and expression for its potential
at any point 32
CONTENTS. Vll
Art. Page
409. The potential of a magnetic shell at any point is the product of
its strength multiplied by the solid angle its boundary sub
tends at the point 32
410. Another method of proof 33
411. The potential at a point on the positive side of a shell of
strength <I> exceeds that on the nearest point on the negative
side by 477$ 34
412. Lamellar distribution of magnetism .. 34
413. Complex lamellar distribution 34
414. Potential of a solenoidal magnet 35
415. Potential of a lamellar magnet 35
416. Vector-potential of a lamellar magnet 36
417. On the solid angle subtended at a given point by a closed curve 36
418. The solid angle expressed by the length of a curve on the sphere 37
419. Solid angle found by two line-integrations 38
420. II expressed as a determinant 39
421. The solid angle is a cyclic function 40
422. Theory of the vector-potential of a closed curve 41
423. Potential energy of a magnetic shell placed in a magnetic field 42
CHAPTER IV.
INDUCED MAGNETIZATION.
424. When a body under the action of magnetic force becomes itself
magnetized the phenomenon is called magnetic induction .. 44
425. Magnetic induction in different substances 45
426. Definition of the coefficient of induced magnetization .. .. 47
427. Mathematical theory of magnetic induction. Poisson's method 47
428. Faraday's method 49
429. Case of a body surrounded by a magnetic medium 51
430. Poisson's physical theory of the cause of induced magnetism .. 53
CHAPTER V.
MAGNETIC PKOBLEMS.
431. Theory of a hollow spherical shell 56
432. Case when K. is large 58
433. When t = l 58
434. Corresponding case in two dimensions. Fig. XV 59
435. Case of a solid sphere, the coefficients of magnetization being
different in different directions 60
viii CONTENTS.
Art. Page
436. The nine coefficients reduced to six. Fig. XVI 61
437. Theory of an ellipsoid acted on by a uniform magnetic force .. 62
438. Cases of very flat and of very long ellipsoids 65
439. Statement of problems solved by Neumann, Kirchhoff and Green 67
440. Method of approximation to a solution of the general problem
when K is very small. Magnetic bodies tend towards places
of most intense magnetic force, and diamagnetic bodies tend
to places of weakest force 69
441. On ship's magnetism 70
CHAPTER VI.
WEBER'S THEORY OF MAGNETIC INDUCTION.
442. Experiments indicating a maximum of magnetization .. .. 74
443. Weber's mathematical theory of temporary magnetization .. 75
444. Modification of the theory to account for residual magnetization 79
445. Explanation of phenomena by the modified theory 81
446. Magnetization, demagnetization, and remagnetization .. .. 83
447. Effects of magnetization on the dimensions of the magnet .. 85
448. Experiments of Joule ' 86
CHAPTER VII.
MAGNETIC MEASUREMENTS.
449. Suspension of the magnet 88
450. Methods of observation by mirror and scale. Photographic
method 89
451. Principle of collimation employed in the Kew magnetometer .. 93
452. Determination of the axis of a magnet and of the direction of
the horizontal component of the magnetic force 94
453. Measurement of the moment of a magnet and of the intensity of
the horizontal component of magnetic force 97
454. Observations of deflexion 99
455. Method of tangents and method of sines 101
456. Observation of vibrations 102
457. Elimination of the effects of magnetic induction 105
458. Statical method of measuring the horizontal force 106
459. Bifilar suspension 107
460. System of observations in an observatory Ill
461. Observation of the dip-circle Ill
CONTENTS. IX
Art. Page
462. J. A. Broun's method of correction 115
463. Joule's suspension 115
464. Balance vertical force magnetometer 117
CHAPTER VIII.
TERRESTRIAL MAGNETISM.
465. Elements of the magnetic force 120
466. Combination of the results of the magnetic survey of a country 121
467. Deduction of the expansion of the magnetic potential of the
earth in spherical harmonics 123
468. Definition of the earth's magnetic poles. They are not at the
extremities of the magnetic axis. False poles. They do not
exist on the earth's surface 123
469. Grauss' calculation of the 24 coefficients of the first four har
monics 124
470. Separation of external from internal causes of magnetic force .. 124
471. The solar and lunar variations 125
472. The periodic variations 125
473. The disturbances and their period of 11 years 126
474. Keflexions on magnetic investigations 126
PART IV.
ELECTROMAGNET ISM.
CHAPTER I.
ELECTROMAGNETIC FORCE.
475. Orsted's discovery of the action of an electric current on a
magnet 128
476. The space near an electric current is a magnetic field .. .. 128
477. Action of a vertical current on a magnet 129
478. Proof that the force due to a straight current of indefinitely
great length varies inversely as the distance 129
479. Electromagnetic measure of the current 130
X CONTENTS.
Art. Page
480. Potential function due to a straight current. It is a function
of many values 130
481. The action of this current compared with that of a magnetic
shell having an infinite straight edge and extending on one
side of this edge to infinity 131
482. A small circuit acts at a great distance like a magnet .. .. 131
483. Deduction from this of the action of a closed circuit of any form
and size on any point not in the current itself 131
484. Comparison between the circuit and a magnetic shell .. .. 132
485. Magnetic potential of a closed circuit 133
486. Conditions of continuous rotation of a magnet about a current 133
487. Form of the magnetic equipotential surfaces due to a closed
circuit. Fig. XVIII 134
488. Mutual action between any system of magnets and a closed
current 135
489. Reaction on the circuit 135
490. Force acting on a wire carrying a current and placed in the
magnetic field 136
491. Theory of electromagnetic rotations .. .. 138
492. Action of one electric circuit on the whole or any portion of
another 139
493. Our method of investigation is that of Faraday 140
494. Illustration of the method applied to parallel currents .. .. 140
495. Dimensions of the unit of current 141
496. The wire is urged from the side on which its magnetic action
strengthens the magnetic force and towards the side on which
it opposes it 141
497. Action of an infinite straight current on any current in its
plane .. • 142
498. Statement of the laws of electromagnetic force. Magnetic force
due to a current 142
499. Generality of these laws .. 143
500. Force acting on a circuit placed in the magnetic field .. ..144
501. Electromagnetic force is a mechanical force acting on the con
ductor, not on the electric current itself 144
CHAPTER II.
MUTUAL ACTION OF ELECTRIC CURRENTS.
502. Ampere's investigation of the law of force between the elements
of electric currents .. 146
CONTENTS. xi
Art. Page
503. His method of experimenting 146
504. Ampere's balance 147
505. Ampere's first experiment. Equal and opposite currents neu
tralize each other 147
506. Second experiment. A crooked conductor is equivalent to a
straight one carrying the same current ..148
507. Third experiment. The action of a closed current as an ele
ment of another current is perpendicular to that element .. 148
508. Fourth experiment. Equal currents in systems geometrically
similar produce equal forces 149
509. In all of these experiments the acting current is a closed one .. 151
510. Both circuits may, however, for mathematical purposes be con
ceived as consisting of elementary portions, and the action
of the circuits as the resultant of the action of these elements 151
511. Necessary form of the relations between two elementary portions
of lines 151
512. The geometrical quantities which determine their relative posi
tion 152
513. Form of the components of their mutual action 153
514. Kesolution of these in three directions, parallel, respectively, to
the line joining them and to the elements themselves .. .. 154
515. General expression for the action of a finite current on the ele
ment of another 154
516. Condition furnished by Ampere's third case of equilibrium .. 155
517. Theory of the directrix and the determinants of electrodynamic
action 156
518. Expression of the determinants in terms of the components
of the vector-potential of the current 157
519. The part of the force which is indeterminate can be expressed
as the space-variation of a potential 157
520. Complete expression for the action between two finite currents 158
521. Mutual potential of two closed currents 158
522. Appropriateness of quaternions in this investigation .. .. 158
523. Determination of the form of the functions by Ampere's fourth
case of equilibrium 159
524. The electrodynamic and electromagnetic units of currents .. 159
525. Final expressions for electromagnetic force between two ele
ments 160
526. Four different admissible forms of the theory 160
527. Of these Ampere's is to be preferred 161
xii CONTENTS.
CHAPTER III.
INDUCTION OF ELECTRIC CUEEENTS.
Art. Page
528. Faraday's discovery. Nature of his methods 162
529. The method of this treatise founded on that of Faraday .. .. 163
530. Phenomena of magneto-electric induction 164
531. General law of induction of currents 166
532. Illustrations of the direction of induced currents .. *. .. 166
533. Induction by the motion of the earth 167
534. The electromotive force due to induction does not depend on
the material of the conductor 168
535. It has no tendency to move the conductor 168
536. Felici's experiments on the laws of induction 168
537. Use of the galvanometer to determine the time-integral of the
electromotive force 170
538. Conjugate positions of two coils 171
539. Mathematical expression for the total current of induction .. 172
540. Faraday's conception of an electrotonic state 173
541. His method of stating the laws of induction with reference to
the lines of magnetic force 174
542. The law of Lenz, and Neumann's theory of induction .. .. 176
543. Helmholtz's deduction of induction from the mechanical action
of currents by the principle of conservation of energy .. .. 176
544. Thomson's application of the same principle 178
545. Weber's contributions to electrical science 178
CHAPTER IV.
INDUCTION OF A CUEEENT ON ITSELF.
546. Shock given by an electromagnet 180
547. Apparent momentum of electricity 180
548. Difference between this case and that of a tube containing a
current of water 181
549. If there is momentum it is not that of the moving electricity .. 181
550. Nevertheless the phenomena are exactly analogous to those of
momentum 181
551. An electric current has energy, which may be called electro-
kinetic energy 182
552. This leads us to form a dynamical theory of electric currents .. 182
CONTENTS. xiii
CHAPTER V.
GENERAL EQUATIONS OF DYNAMICS.
Art. Page
553. Lagrange's method furnishes appropriate ideas for the study of
the higher dynamical sciences 184
554. These ideas must be translated from mathematical into dy
namical language 184
555. Degrees of freedom of a connected system 185
556. Generalized meaning of velocity 186
557. Generalized meaning of force , .. ..186
558. Generalized meaning of momentum and impulse ,. ,. .. 186
559. Work done by a small impulse .. ., 187
560. Kinetic energy in terms of momenta, (Tp) .. .. ,. .. 188
561. Hamilton's equations of motion .. .. , 189
562. Kinetic energy in terms of the velocities and momenta, (Tp,j) .. 190
563. Kinetic energy in terms of velocities, (T^) ,, ., .. .. 191
564. Relations between Tp and T^, p and q 191
565. Moments and products of inertia and mobility .. .. ,. 192
566. Necessary conditions which these coefficients must satisfy .. 193
567. Relation between mathematical, dynamical, and electrical ideas 193
CHAPTER VI.
APPLICATION OF DYNAMICS TO ELECTROMAGNETISM.
568. The electric current possesses energy 195
569. The current is a kinetic phenomenon 195
570. Work done by electromotive force 196
571. The most general expression for the kinetic energy of a system
including electric currents ., .. .. 197
572. The electrical variables do not appear in this expression .. .. 198
573. Mechanical force acting on a conductor 198
574. The part depending on products of ordinary velocities and
strengths of currents does not exist 200
575. Another experimental test , ,, ., .. 202
576. Discussion of the electromotive force 204
577. If terms involving products of velocities and currents existed
they would introduce electromotive forces, which are not ob
served ,. ,. ,. 204
CHAPTER VII.
ELECTROKINETICS.
578. The electrokinetic energy of a system of linear circuits .. .. 206
579. Electromotive force in each circuit . . 207
xiv CONTENTS.
Art. Page
580. Electromagnetic force 208
581. Case of two circuits 208
582. Theory of induced currents 209
583. Mechanical action between the circuits 210
584. All the phenomena of the mutual action of two circuits depend
on a single quantity, the potential of the two circuits .. .. 210
CHAPTER VIII.
EXPLOBATION OF THE FIELD BY MEANS OF THE SECONDARY CIRCUIT.
585. The electrokinetic momentum of the secondary circuit .. .. 211
586. Expressed as a line-integral 211
587. Any system of contiguous circuits is equivalent to the circuit
formed by their exterior boundary 212
588. Electrokinetic momentum expressed as a surface -integral .. .212
589. A crooked portion of a circuit equivalent to a straight portion 213
590. Electrokinetic momentum at a point expressed as a vector, Ql .. 214
591. Its relation to the magnetic induction, 3B. Equations (A) .. 214
592. Justification of these names 215
593. Conventions with respect to the signs of translations and rota
tions 216
594. Theory of a sliding piece 217
595. Electromotive force due to the motion of a conductor .. .. 218
596. Electromagnetic force on the sliding piece ..218
597. Four definitions of a line of magnetic induction 219
598. General equations of electromotive force, (B) 219
599. Analysis of the electromotive force 222
600. The general equations referred to moving axes 223
601. The motion of the axes changes nothing but the apparent value
of the electric potential 224
602. Electromagnetic force on a conductor 224
603. Electromagnetic force on an element of a conducting body.
Equations (C) 226
CHAPTER IX.
GENERAL EQUATIONS.
604. Recapitulation 227
605. Equations of magnetization, (D) 228
606. Relation between magnetic force and electric currents .. •• 229
607. Equations of electric currents, (E) 230
608. Equations of electric displacement, (F) 232
CONTENTS. xv
Art. Page
609. Equations of electric conductivity, (G) 232
610. Equations of total currents, (H) 232
611. Currents in terms of electromotive force, (I) .. .. .. .. 233
612. Volume-density of free electricity, (J) 233
613. Surface-density of free electricity, (K) 233
614. Equations of magnetic permeability, (L) 233
615. Ampere's theory of magnets 234
616. Electric currents in terms of electrokinetic momentum .. .. 234
617. Vector-potential of electric currents 236
618. Quaternion expressions for electromagnetic quantities .. .. 236
619. Quaternion equations of the electromagnetic field 237
CHAPTER X.
DIMENSIONS OF ELECTKIC UNITS.
620. Two systems of units .. .. 239
621. The twelve primary quantities 239
622. Fifteen relations among these quantities 240
623. Dimensions in terms of [e] and [m] 241
624. Reciprocal properties of the two systems 241
625. The electrostatic and the electromagnetic systems 241
626. Dimensions of the 12 quantities in the two systems .. .. 242
627. The six derived units 243
628. The ratio of the corresponding units in the two systems .. 243
629. Practical system of electric units. Table of practical units .. 244
CHAPTER XI.
ENERGY AND STRESS.
630. The electrostatic energy expressed in terms of the free electri
city and the potential 246
631. The electrostatic energy expressed in terms of the electromotive
force and the electric displacement 246
632. Magnetic energy in terms of magnetization and magnetic force 247
633. Magnetic energy in terms of the square of the magnetic force .. 247
634. Electrokinetic energy in terms of electric momentum and electric
current 248
635. Electrokinetic energy in terms of magnetic induction and mag
netic force 248
636. Method of this treatise 249
637. Magnetic energy and electrokinetic energy compared .. .. 249
638. Magnetic energy reduced to electrokinetic energy 250
xvi CONTENTS.
Art. Page
639. The force acting on a particle of a substance due to its magnet
ization 251
640. Electromagnetic force due to an electric current passing through
it 252
641. Explanation of these forces by the hypothesis of stress in a
medium 253
642. General character of the stress required to produce the pheno
mena 255
643. When there is no magnetization the stress is a tension in the
direction of the lines of magnetic force, combined with a
pressure in all directions at right angles to these lines, the
magnitude of the tension and pressure being — — • ^2, where •$
O7T
is the magnetic force 256
644. Force acting on a conductor carrying a current 257
645. Theory of stress in a medium as stated by Faraday .. .. 257
646. Numerical value of magnetic tension 258
CHAPTER XII.
CURRENT-SHEETS.
647. Definition of a current-sheet 259
648. Current-function 259
649. Electric potential , 260
650. Theory of steady currents 260
651. Case of uniform conductivity 260
652. Magnetic action of a current-sheet with closed currents .. .. 261
653. Magnetic potential due to a current-sheet 262
654. Induction of currents in a sheet of infinite conductivity .. .. 262
655. Such a sheet is impervious to magnetic action 263
656. Theory of a plane current-sheet 263
657. The magnetic functions expressed as derivatives of a single
function 264
658. Action of a variable magnetic system on the sheet 266
659. When there is no external action the currents decay, and their
magnetic action diminishes as if the sheet had moved off with
constant velocity R 267
660. The currents, excited by the instantaneous introduction of a
magnetic system, produce an effect equivalent to an image of
that system 267
661. This image moves away from its original position with velo
city R 268
662. Trail of images formed by a magnetic system in continuous
motion . 268
CONTENTS. xvn
Art. Page
663. Mathematical expression for the effect of the induced currents 269
664. Case of the uniform motion of a magnetic pole 269
665. Value of the force acting on the magnetic pole 270
666. Case of curvilinear motion 271
667. Case of motion near the edge of the sheet .. .. ..-'.', 271
668. Theory of Arago's rotating disk 271
669. Trail of images in the form of a helix 274
670. Spherical current-sheets 275
671. The vector- potential 276
672. To produce a field of constant magnetic force within a spherical
shell 277
673. To produce a constant force on a suspended coil 278
674. Currents parallel to a plane 278
675. A plane electric circuit. A spherical shell. An ellipsoidal
shell 279
676. A solenoid 280
677. A long solenoid 281
678. Force near the ends 282
679. A pair of induction coils 282
680. Proper thickness of wire 283
G81. An endless solenoid 284
CHAPTER XIII.
PAKALLEL CURRENTS.
682. Cylindrical conductors 286
683. The external magnetic action of a cylindric wire depends only
on the whole current through it .. 287
684. The vector-potential 288
685. Kinetic energy of the current 288
686. Repulsion between the direct and the return current .. .. 289
687. Tension of the wires. Ampere's experiment ,. 289
688. Self-induction of a wire doubled on itself 290
689. Currents of varying intensity in a cylindric wire 291
690. Relation between the electromotive force and the total current 292
691. Geometrical mean distance of two figures in a plane .. ,. 294
692. Particular cases 294
693. Application of the method to a coil of insulated wires .. .. 296
CHAPTER XIV.
CIRCULAR CURRENTS.
694. Potential due to a spherical bowl 299
695. Solid angle subtended by a circle at any point 301
VOL. II. b
xviii CONTENTS.
Art. Page
696. Potential energy of two circular currents 302
697. Moment of the couple acting between two coils 303
698. Values of Q? 303
699. Attraction between two parallel circular currents 304
700. Calculation of the coefficients for a coil of finite section .. .. 304
701. Potential of two parallel circles expressed by elliptic integrals 305
702. Lines of force round a circular current. Fig. XVIII .. .. 307
703. Differential equation of the potential of two circles 307
704. Approximation when the circles are very near one another .. 309
705. Further approximation 310
706. Coil of maximum self-induction 311
CHAPTER XV.
ELECTROMAGNETIC INSTRUMENTS.
707. Standard galvanometers and sensitive galvanometers .. .. 313
708. Construction of a standard coil 314
709. Mathematical theory of the galvanometer 315
710. Principle of the tangent galvanometer and the sine galvano
meter 316
711. Galvanometer with a single coil 316
712. Gaugain's eccentric suspension 317
713. Helmholtz's double coil. Fig. XIX 318
714. Galvanometer with four coils 319
715. Galvanometer with three coils 319
716. Proper thickness of the wire of a galvanometer 321
717. Sensitive galvanometers 322
718. Theory of the galvanometer of greatest sensibility 322
719. Law of thickness of the wire 323
720. Galvanometer with wire of uniform thickness 325
721. Suspended coils. Mode of suspension 326
722. Thomson's sensitive coil 326
723. Determination of magnetic force by means of suspended coil
and tangent galvanometer 327
724. Thomson's suspended coil and galvanometer combined .. .. 328
725. Weber's electrodynamometer 328
726. Joule's current -weigher 332"
727. Suction of solenoids 333
728. Uniform force normal to suspended coil 333
729. Electrodynamometer with torsion-arm 334
CONTENTS. xix
CHAPTER XVI.
ELECTROMAGNETIC OBSERVATIONS.
Art. Page
730. Observation of vibrations , ;. 335
731. Motion in a logarithmic spiral 336
732. Eectilinear oscillations in a resisting medium 337
733. Values of successive elongations 338
734. Data and qusesita 338
735. Position of equilibrium determined from three successive elon
gations 338
736. Determination of the logarithmic decrement 339
737. When to stop the experiment 339
738. Determination of the time of vibration from three transits .. 339
739. Two series of observations 340
740. Correction for amplitude and for damping 341
741. Dead beat galvanometer 341
742. To measure a constant current with the galvanometer .. .. 342
743. Best angle of deflexion of a tangent galvanometer 343
744. Best method of introducing the current 343
745. Measurement of a current by the first elongation 344
746. To make a series of observations on a constant current .. .. 345
747. Method of multiplication for feeble currents 345
748. Measurement of a transient current by first elongation .. .. 346
749. Correction for damping 347
750. Series of observations. Zurilckwerfungs methode 348
751. Method of multiplication 350
CHAPTER XVII.
ELECTRICAL MEASUREMENT OF COEFFICIENTS OF INDUCTION.
752. Electrical measurement sometimes more accurate than direct
measurement 352
753. Determination of G^ 353
754. Determination of gl 354
755. Determination of the mutual induction of two coils .. .. 354
756. Determination of the self-induction of a coil 356
757. Comparison of the self-induction of two coils 357
CHAPTER XVIII.
DETERMINATION OF RESISTANCE IN ELECTROMAGNETIC MEASURE.
758. Definition of resistance 358
759. Kirchhoff's method 358
XX CONTENTS.
Art. Page
760. Weber's method by transient currents 360
761. His method of observation 361
762. Weber's method by damping 361
763. Thomson's method by a revolving coil 364
764. Mathematical theory of the revolving coil ..- 364
765. Calculation of the resistance 365
766. Corrections 366
767. Joule's calorimetric method 367
CHAPTER XIX.
COMPARISON OF ELECTROSTATIC WITH ELECTROMAGNETIC UNITS.
768. Nature and importance of the investigation 368
769. The ratio of the units is a velocity 369
770. Current by convection 370
771. Weber and Kohlrausch's method 370
772. Thomson's method by separate electrometer and electrodyna-
mometer 372
773. Maxwell's method by combined electrometer and electrodyna-
mometer 372
774. Electromagnetic measurement of the capacity of a condenser.
Jenkin's method 373
775. Method by an intermittent current 374
776. Condenser and Wippe as an arm of Wheatstone's bridge .. 375
777. Correction when the action is too rapid 376
778. Capacity of a condenser compared with the self-induction of a
coil 377
779. Coil and condenser combined 379
780. Electrostatic measure of resistance compared with its electro
magnetic measure 382
CHAPTER XX.
ELECTROMAGNETIC THEORY OF LIGHT.
781. Comparison of the properties of the electromagnetic medium
with those of the medium in the undulatory theory of light 383
782. Energy of light during its propagation 384
783. Equation of propagation of an electromagnetic disturbance .. 384
784. Solution when the medium is a non-conductor 386
785. Characteristics of wave-propagation 386
786. Velocity of propagation of electromagnetic disturbances .. .. 387
787. Comparison of this velocity with that of light 387
CONTENTS. xxi
Art. Page
788. The specific inductive capacity of a dielectric is the square of
its index of refraction 388
789. Comparison of these quantities in the case of paraffin .. .. 388
790. Theory of plane waves 389
791. The electric displacement and the magnetic disturbance are in
the plane* of the wave-front, and perpendicular to each other 390
792. Energy and stress during radiation 391
793. Pressure exerted by light .. .. 391
794. Equations of motion in a crystallized medium 392
795. Propagation of plane waves •,. .. 393
796. Only two waves are propagated 393
797. The theory agrees with that of Fresnel 394
798. Relation between electric conductivity and opacity .. .. 394
799. Comparison with facts 395
800. Transparent metals 395
801. Solution of the equations when the medium is a conductor .. 395
802. Case of an infinite medium, the initial state being given .. 396
803. Characteristics of diffusion 397
804. Disturbance of the electromagnetic field when a current begins
to flow 397
805. Rapid approximation to an ultimate state 398
CHAPTER XXI.
MAGNETIC ACTION ON LIGHT.
806. Possible forms of the relation between magnetism and light .. 399
807. The rotation of the plane of polarization by magnetic action .. 400
808. The laws of the phenomena 400
809. Verdet's discovery of negative rotation in ferromagnetic media 400
810. Rotation produced by quartz, turpentine, &c., independently of
magnetism 401
811. Kinematical analysis of the phenomena 402
812. The velocity of a circularly-polarized ray is different according
to its direction of rotation , 402
813. Right and left-handed rays 403
814. In media which of themselves have the rotatory property the
velocity is different for right and left-handed configurations 403
815. In media acted on by magnetism the velocity is different for
opposite directions of rotation 404
816. The luminiferous disturbance, mathematically considered, is a
vector 404
817. Kinematic equations of circularly-polarized light 405
xxii CONTENTS.
Art. Page
818. Kinetic and potential energy of the medium 406
819. Condition of wave-propagation 406
820. The action of magnetism must depend on a real rotation about
the direction of the magnetic force as an axis 407
821. Statement of the results of the analysis of the phenomenon .. 407
822. Hypothesis of molecular vortices 408
823. Variation of the vortices according to Helmholtz's law .. .. 409
824. Variation of the kinetic energy in the disturbed medium .. 409
825.- Expression in terms of the current and the velocity .. .. 410
826. The kinetic energy in the case of plane waves 410
827. The equations of motion 411
828. Velocity of a circularly-polarized ray 411
829. The magnetic rotation 412
830. Researches of Verdet 413
831. Note on a mechanical theory of molecular vortices 415
CHAPTER XXII.
ELECTRIC THEOEY OF MAGNETISM.
832. Magnetism is a phenomenon of molecules 418
833. The phenomena of magnetic molecules may be imitated by
electric currents 419
834. Difference between the elementary theory of continuous magnets
and the theory of molecular currents 419
835. Simplicity of the electric theory 420
836. Theory of a current in a perfectly conducting circuit .. .. 420
837. Case in which the current is entirely due to induction .. .. 421
838. Weber's theory of diamagnetism 421
839. Magnecrystallic induction 422
840. Theory of a perfect conductor 422
841. A medium containing perfectly conducting spherical molecules 423
842. Mechanical action of magnetic force on the current which it
excites 423
843. Theory of a molecule with a primitive current 424
844. Modifications of Weber's theory 425
845. Consequences of the theory 425
CHAPTER XXIII.
THEORIES OF ACTION AT A DISTANCE.
846. Quantities which enter into Ampere's formula 426
847. Relative motion of two electric particles 426
CONTENTS. xxiii
Art. Page
848. Relative motion of four electric particles. Fechner's theory .. 427
849. Two new forms of Ampere's formula 428
850. Two different expressions for the force between two electric
particles in motion 428
851. These are due to Gauss and to Weber respectively 429
852. All forces must be consistent with the principle of the con
servation of energy 429
853. Weber's formula is consistent with this principle but that of
Gauss is not 429
854. Helmholtz's deductions from Weber's formula 430
855. Potential of two currents 431
856. Weber's theory of the induction of electric currents .. .. 431
857. Segregating force in a conductor 432
858. Case of moving conductors 433
859. The formula of Gauss leads to an erroneous result 434
860. That of Weber agrees with the phenomena 434
861. Letter of Gauss to Weber 435
862. Theory of Riemann 435
863. Theory of C. Neumann 435
864. Theory of Betti 436
865. Repugnance to the idea of a medium 437
866. The idea of a medium cannot be got rid of 437
ERRATA. VOL. II.
p. 11, 1.1, for r.
dV, d2 .lx
read W = m9-^— = — m, m,^- — — (-)•
2 2^'
„ equation (8), insert — before each side of this equation.
p. 1 3, last line but one, dele — .
p. 14, 1. 8, for XVII read XIV.
p. 15, equation (5), for VpdS read Vpdxdydz.
p. 16, 1. 4 from bottom, after equation (3) insert of Art. 389.
p. 17, equation (14), for r read r5.
p. 21, 1. 1, for 386 read 385.
„ 1. 7 from bottom for in read on.
p. 28, last line but one, for 386 read 385.
dF dH _ <W d#
p. 41, equation (10), for ^--^ ttffi ^-^'
p. 43, equation (14), put accents on #, ?/, z.
p. 50, equation (19), for — , &c. rmc? — , &c., inverting all the differ
du x cL v
ential coefficients.
p. 51, 1. 11, for 309 read 310.
p. 61, 1. 16, for Y=Fsm0 read Z=Fsm6.
„ equation (10), for TT read 7i2.
p. 62, equation (13), for § read f.
p. 63, 1. 3, for pdr read pdv.
p. 67, right-hand side of equation should be
4
p. 120, equation (1), for downwards read upwards.
„ equation (2), insert — before the right-hand member of each
equation.
p. 153, 1. 15, for =(3 read =/3'.
p. 155, 1. 8, for A A read AP.
p. 190, equation (11), for Fbq1 read Fb^.
p. 192, 1. 22, for Tp read Tp.
p. 193, after 1. 5 from bottom, insert, But they will be all satisfied pro
vided the n determinants formed by the coefficients having the
indices 1 ; 1, 2 ; 1, 2, 3, &c. ; 1, 2, 3, ..n are none of them
negative.
p. 197, 1. 22, for (x^ #15 &c.) read fax^&c.
„ 1. 23, for (xlt 052, &c.) read (x-^x^)^ &c.
p. 208, 1. 2 from bottom, for Ny£ read \Ny£.
p. 222, 1. 9 from bottom, for -^~ or % read -^ or -&
p. 235, equations (5), for - read ju j and in (6) for — read — •
p. 245, first number of last column in the table should be 1010.
p. 258, 1. 14, for perpendicular to read along.
p. 265, 1. 2 after equation (9), for -~ read -=~«
ay ciy
ERRATA. VOL. II.
3 from bottom, for (-) read (-) -
p. ;281y equation (19), for n read %.
p. 282, 1. 8, for z2 read z*.
p. 289, equation (22), for 4a24 read 2af ; and for 4«'24 read 2 a'
p. 293, equation (17), dele — .
p. 300, 1. 7, for when read where.
„ 1. 17, insert — after =.
„ 1. 26, for Q* read ft.
p. 301, equation (4') for / read r\
„ equation (5), insert — after = .
p. 302, 1. 4 from bottom, for M= \ read M=—J-
„ 1. 3 from bottom, insert at the beginning M—
n the denominator of the last term should be c,
„ last line, before the first bracket, for c22 read c2.
p. 303, 1. 1 1 from bottom, for ft' read ft',
p. 306, 1. 14, for 277 read 4-77.
„ 1. 15, for >fAa read 2 V~Aa.
1. 19 should be
7 Tlf
„ lines 23 and 27, change the sign of --=— •
p. 316, equation (3), for =My- read my.
p. 317, 1. 7, for ~| read -3.
p. 318, 1. 8 from bottom for 36 to 31 read ^36 to
p. 320, 1. 9, for 627, read 672.
„ last line, after = insert f.
p. 324, equation (14) should be - ~ (1 -H—y^)=~^ = constant.
TT y y
p. 325, 1. 5 from bottom, should be #=| ^-2 ^ (a^-a3).
p. 346, 1. 2, for 0 read 0^
p. 359, equation (2), /or ^^ read —Ex.
p. 365, equation (3), last term, dele y.
PART III.
MAGNETISM.
CHAPTEK I.
ELEMENTARY THEORY OF MAGNETISM.
371.] CERTAIN bodies, as, for instance, the iron ore called load
stone, the earth itself, and pieces of steel which have been sub
jected to certain treatment, are found to possess the following
properties, and are called Magnets. •
If, near any part of the earth's surface except the Magnetic
Poles, a magnet be suspended so as to turn freely about a vertical
axis, it will in general tend to set itself in a certain azimuth, and
if disturbed from this position it will oscillate about if. An un-
magnetized body has no such tendency, but is in equilibrium in
all azimuths alike.
372.] It is found that the force which acts on the body tends
to cause a certain line in the body, called the Axis of the Magnet,
to become parallel to a certain line in space, called the Direction
of the Magnetic Force.
Let us suppose the magnet suspended so as to be free to turn
in all directions about a fixed point. To eliminate the action of
its weight we may suppose this point to be its centre of gravity.
Let it come to a position^of equilibrium. Mark two points on
the magnet, and note their positions in space. Then let the
magnet be placed in a new position of equilibrium, and note the
positions in space of the two marked points on the magnet.
Since the axis of the magnet coincides with the direction of
magnetic force in both positions, we have to find that line in
the magnet which occupies the same position in space before and
VOL. II. B
2 ELEMENTARY THEORY OF MAGNETISM. [373-
after the motion. It appears, from the theory of the motion of
>•;{ ^'bodies of invariable form, that such a line always exists, and that
a motion equivalent to the actual motion might have taken place
by simple rotation round this line.
To find the line, join the first and last positions of each of the
marked points, and draw planes bisecting these lines at right
angles. The intersection of these planes will be the line required,
which indicates the direction of the axis of the magnet and the
direction of the magnetic force in space.
The method just described is not convenient for the practical
determination of these directions. We shall return to this subject
when we treat of Magnetic Measurements.
The direction of the magnetic force is found to be different at
different parts of the earth's surface. If the end of the axis of
the magnet which points in a northerly direction be marked, it
has been found that the direction in which it sets itself in general
deviates from the true meridian to a considerable extent, and that
the marked end points on the whole downwards in the northern
fc hemisphere and upwards in the southern.
The azimuth of the direction of the magnetic force, measured
from the true north in a westerly direction, is called the Variation,
or the Magnetic Declination. The angle between the direction of
the magnetic force and the horizontal plane is called the Magnetic
Dip. These two angles determine the direction of the magnetic
force, and, when the magnetic intensity is also known, the magnetic
force is completely determined. The determination of the values
of these three elements at different parts of the earth's surface,
the discussion of the manner in which they vary according to the
place and time of observation, and the investigation of the causes
of the magnetic force and its variations, constitute the science of
Terrestrial Magnetism.
373.] Let us now suppose that the axes of several magnets have
been determined, and the end of each which points north marked.
Then, if one of these be freely suspended and another brought
near it, it is found that two marked ends repel each other, that
a marked and an unmarked end attract each other, and that two
unmarked ends repel each other.
If the magnets are in the form of long rods or wires, uniformly
and longitudinally magnetized, see below, Art. 384, it is found
that the greatest manifestation of force occurs when the end of
one magnet is held near the end of the other, and that the
374-] LAW OF MAGNETIC FORCE. 3
phenomena can be accounted for by supposing- that like ends of
the magnets repel each other, that unlike ends attract each other,
and that the intermediate parts of the magnets have no sensible
mutual action.
The ends of a long thin magnet are commonly called its Poles.
In the case of an indefinitely thin magnet, uniformly magnetized
throughout its length, the extremities act as centres of force, and
the rest of the magnet appears devoid of magnetic action. In
all actual magnets the magnetization deviates from uniformity, so
that no single points can be taken as the poles. Coulomb, how
ever, by using long thin rods magnetized with care, succeeded in
establishing the law of force between two magnetic poles *.
The repulsion between two magnetic poles is in the straight line joining
them, and is numerically equal to the product of the strengths of
the poles divided by the square of the distance between them.
374.] This law, of course, assumes that the strength of each
pole is measured in terms of a certain unit, the magnitude of which
may be deduced from the terms of the law.
The unit-pole is a pole which points north, and is such that,
when placed at unit distance from another unit-pole, it repels it
with unit offeree, the unit of force being defined as in Art. 6. A
pole which points south is reckoned negative.
If m1 and m2 are the strengths of two magnetic poles, I the
distance between them, and / the force of repulsion, all expressed
numerically, then .
~
But if [m], [I/I and [F] be the concrete units of magnetic pole,
length and force, then
whence it follows that
or [m] = \Il*T-lM*\.
The dimensions of the unit pole are therefore f as regards length,
( — 1) as regards time, and \ as regards mass. These dimensions
are the same as those of the electrostatic unit of electricity, which
is specified in exactly the same way in Arts. 41, 42.
* His experiments on magnetism with the Torsion Balance are contained in
the Memoirs of the Academy of Paris, 1780-9, and in Biot's Traite de Physique,
torn. iii.
4 ELEMENTARY THEORY OF MAGNETISM. [375-
375.] The accuracy of this law may be considered to have
been established by the experiments of Coulomb with the Torsion
Balance, and confirmed by the experiments of Gauss and Weber,
and of all observers in magnetic observatories, who are every day
making measurements of magnetic quantities, and who obtain results
which would be inconsistent with each other if the law of force
had been erroneously assumed. It derives additional support from
its consistency with the laws of electromagnetic phenomena.
376.] The quantity which we have hitherto called the strength
of a pole may also be called a quantity of ' Magnetism,' provided
we attribute no properties to ' Magnetism ' except those observed
in the poles of magnets.
Since the expression of the law of force between given quantities
of 'Magnetism' has exactly the same mathematical form as the
law of force between quantities of 'Electricity' of equal numerical
value, much of the mathematical treatment of magnetism must be
similar to that of electricity. There are, however, other properties
of magnets which must be borne in mind, and which may throw
some light on the electrical properties of bodies.
Relation between the Poles of a Magnet.
377.] The quantity of magnetism at one pole of a magnet is
always equal and opposite to that at the other, or more generally
thus : —
In every Magnet the total quantity of Magnetism (reckoned alge
braically) is zero.
Hence in a field of force which is uniform and parallel throughout
the space occupied by the magnet, the force acting on the marked
end of the magnet is exactly equal, opposite and parallel to that on
the unmarked end, so that the resultant of the forces is a statical
couple, tending to place the axis of the magnet in a determinate
direction, but not to move the magnet as a whole in any direction.
This may be easily proved by putting the magnet into a small
vessel and floating it in water. The vessel will turn in a certain
direction, so as to bring the axis of the magnet as near as possible
to the direction of the earth's magnetic force, but there will be no
motion of the vessel as a whole in any direction ; so that there can
be no excess of the force towards the north over that towards the
south, or the reverse. It may also be shewn from the fact that
magnetizing a piece of steel does not alter its weight. It does alter
the apparent position of its centre of gravity, causing it in these
380.] MAGNETIC 'MATTER/ 5
latitudes to shift along the axis towards the north. The centre
of inertia, as determined by the phenomena of rotation, remains
unaltered.
378.] If the middle of a long thin magnet be examined, it is
found to possess no magnetic properties, but if the magnet be
broken at that point, each of the pieces is found to have a magnetic
pole at the place of fracture, and this new pole is exactly equal
and opposite to the other pole belonging to that piece. It is
impossible, either by magnetization, or by breaking magnets, or
by any other means, to procure a magnet whose poles are un
equal.
If we break the long thin magnet into a number of short pieces
we shall obtain a series of short magnets, each of which has poles
of nearly the same strength as those of the original long magnet.
This multiplication of poles is not necessarily a creation of energy,
for we must remember that after breaking the magnet we have to
do work to separate the parts, in consequence of their attraction
for one another.
379.] Let us now put all the pieces of the magnet together
as at first. At each point of junction there will be two poles
exactly equal and of opposite kinds, placed in contact, so that their
united action on any other pole will be null. The magnet, thus
rebuilt, has therefore the same properties as at first, namely two
poles, one at each end, equal and opposite to each other, and the
part between these poles exhibits no magnetic action.
Since, in this case, we know the long magnet to be made up
of little short magnets, and since the phenomena are the same
as in the case of the unbroken magnet, we may regard the magnet,
even before being broken, as made up of small particles, each of
which has two equal and opposite poles. If we suppose all magnets
to be made up of such particles, it is evident that since the
algebraical quantity of magnetism in each particle is zero, the
quantity in the whole magnet will also be zero, or in other words,
its poles will be of equal strength but of opposite kind.
Theory of Magnetic ''Matter?
380.] Since the form of the law of magnetic action is identical
with that of electric action, the same reasons which can be given
for attributing electric phenomena to the action of one ' flu id'
or two ' fluids' can also be used in favour of the existence of a
magnetic matter, or of two kinds of magnetic matter, fluid or
6 ELEMENTARY THEORY OF MAGNETISM. [380.
otherwise. In fact, a theory of magnetic matter, if used in a
purely mathematical sense, cannot fail to explain the phenomena,
provided new laws are freely introduced to account for the actual
facts.
One of these new laws must be that the magnetic fluids cannot
pass from one molecule or particle of the magnet to another, but
that the process of magnetization consists in separating to a certain
extent the two fluids within each particle, and causing the one fluid
to be more concentrated at one end, and the other fluid to be more
concentrated at the other end of the particle. This is the theory of
Poisson.
A particle of a magnetizable body is, on this theory, analogous
to a small insulated conductor without charge, which on the two-
fluid theory contains indefinitely large but exactly equal quantities
of the two electricities. When an electromotive force acts on the
conductor, it separates the electricities, causing them to become
manifest at opposite sides of the conductor. In a similar manner,
according to this theory, the magnetizing force causes the two
kinds of magnetism, which were originally in a neutralized state,
to be separated, and to appear at opposite sides of the magnetized
particle.
In certain substances, such as soft iron and those magnetic
substances which cannot be permanently magnetized, this magnetic
condition, like the electrification of the conductor, disappears when
the inducing force is removed. In other substances, such as hard
steel, the magnetic condition is produced with difficulty, and, when
produced, remains after the removal of the inducing force.
This is expressed by saying that in the latter case there is a
Coercive Force, tending to prevent alteration in the magnetization,
which must be overcome before the power of a magnet can be
either increased or diminished. In the case of the electrified body
this would correspond to a kind of electric resistance, which, unlike
the resistance observed in metals, would be equivalent to complete
insulation for electromotive forces below a certain value.
This theory of magnetism, like the corresponding theory of
electricity, is evidently too large for the facts, and requires to be
restricted by artificial conditions. For it not only gives no reason
why one body may not differ from another on account of having
more of both fluids, but it enables us to say what would be the
properties of a body containing an excess of one magnetic fluid.
It is true that a reason is given why such a body cannot exist,
381.] MAGNETIC POLARIZATION. 7
but this reason is only introduced as an after-thought to explain
this particular fact. It does not grow out of the theory.
381.] We must therefore seek for a mode of expression which
shall not be capable of expressing too much, and which shall leave
room for the introduction of new ideas as these are developed from
new facts. This, I think, we shall obtain if we begin by saying
that the particles of a magnet are Polarized.
Meaning of the term ' Polarization?
When a particle of a body possesses properties related to a
certain line or direction in the body, and when the body, retaining
these properties, is turned so that this direction is reversed, then
if as regards other bodies these properties of the particle are
reversed, the particle, in reference to these properties, is said to be
polarized, and the properties are said to constitute a particular
kind of polarization.
Thus we may say that the rotation of a body about an axis
constitutes a kind of polarization, because if, while the rotation
continues, the direction of the axis is turned end for end, the body
will be rotating in the opposite direction as regards space.
A conducting particle through which there is a current of elec
tricity may be said to be polarized, because if it were turned round,
and if the current continued to flow in the same direction as regards
the particle, its direction in space would be reversed.
In short, if any mathematical or physical quantity is of the
nature of a vector, as defined in Art. 11, then any body or particle
to which this directed quantity or vector belongs may be said to
be Polarized *9 because it has opposite properties in the two opposite
directions or poles of the directed quantity.
The poles of the earth, for example, have reference to its rotation,
and have accordingly different names.
* The word Polarization has been used in a sense not consistent with this in
Optics, where a ray of light is said to be polarized when it has properties relating
to its sides, which are identical on opposite sides of the ray. This kind of polarization
refers to another kind of Directed Quantity, which may be called a Dipolar Quantity,
in opposition to the former kind, which may be called Unipolar.
When a dipolar quantity is turned end for end it remains the same as before.
Tensions and Pressures in solid bodies, Extensions, Compressions and Distortions
and most of the optical, electrical, and magnetic properties of crystallized bodies
are dipolar quantities.
The property produced by magnetism in transparent bodies of twisting the plane
of polarization of the incident light, is, like magnetism itself, a unipolar property.
The rotatory property referred to in Art. 303 is also unipolar.
8 ELEMENTARY THEORY OF MAGNETISM. [382.
Meaning of the term ' Magnetic Polarization.''
382.] In speaking of the state of the particles of a magnet as
magnetic polarization, we imply that each of the smallest parts
into which a magnet may be divided has certain properties related
to a definite direction through the particle, called its Axis of
Magnetization, and that the properties related to one end of this
axis are opposite to the properties related to the other end.
The properties which we attribute to the particle are of the same
kind as those which we observe in the complete magnet, and in
assuming that the particles possess these properties, we only assert
what we can prove by breaking the magnet up into small pieces,
for each of these is found to be a magnet.
Properties of a Magnetized Particle.
383.] Let the element dxdydz be a particle of a magnet, and
let us assume that its magnetic properties are those of a magnet
the strength of whose positive pole is mt and whose length is ds.
Then if P is any point in space distant r from the positive pole and
/ from the negative pole, the magnetic potential at P will be
— due to the positive pole, and -- -^ due to the negative pole, or
If ds, the distance between the poles, is very small, we may put
/— r = dscos e, (2)
where e is the angle between the vector drawn from the magnet
to P and the axis of the magnet, or
, N
cose. (3)
Magnetic Moment.
384.] The product of the length of a* uniformly and longitud
inally magnetized bar magnet into the strength of its positive pole
is called its Magnetic Moment.
Intensity of Magnetization.
The intensity of magnetization of a magnetic particle is the ratio
of its magnetic moment to its volume. We shall denote it by /.
The magnetization at any point of a magnet may be defined
by its intensity and its direction. Its direction may be defined by
its direction-cosines A, /u,, v.
385.] COMPONENTS OF MAGNETIZATION. 9
Components of Magnetization.
The magnetization at a point of a magnet (being a vector or
directed quantity) may be expressed in terms of its three com
ponents referred to the axes of coordinates. Calling these A, B, C,
A = I\, B = Iy., C=Iv,
and the numerical value of I is given by the equation (4)
ja = A*+B* + C2. (5)
385.] If the portion of the magnet which we consider is the
differential element of volume dxdydz, and if / denotes the intensity
of magnetization of this element, its magnetic moment is Idxdydz.
Substituting this for mds in equation (3), and remembering that
rcose = \(£-x)+iL(ri—y) + v(C—z), (6)
where £, 77, f are the coordinates of the extremity of the vector r
drawn from the point (#, y, z), we find for the potential at the point
(£, 77, () due to the magnetized element at (a?, y, z\
W= {A(£-x) + B(ri-y)+C({-z)}±;dxdydz. (7)
To obtain the potential at the point (£. r], f) due to a magnet of
finite dimensions, we must find the integral of this expression for
every element of volume included within the space occupied by
the magnet, or
(8)
Integrating by parts, this becomes
dc
where the double integration in the first three terms refers to the
surface of the magnet, and the triple integration in the fourth to
the space within it.
If I, m, n denote the direction-cosines of the normal drawn
outwards from the element of surface dS, we may write, as in
Art. 21 j the sum of the first three terms,
where the integration is to be extended over the whole surface of
the magnet.
10 ELEMENTARY THEORY OF MAGNETISM. [386.
If we now introduce two new symbols a and p} defined by the
equations <r =
(dA dB dC^
p: ~^ + ^ + ^;j
the expression for the potential may be written
386.] This expression is identical with that for the electric
potential due to a body on the surface of which there is an elec
trification whose surface-density is o-, while throughout its substance
there is a bodily electrification whose volume-density is p. Hence,
if we assume cr and p to be the surface- and volume-densities of the
distribution of an imaginary substance, which we have called
t magnetic matter,' the potential due to this imaginary distribution
will be identical with that due to the actual magnetization of every
element of the magnet.
The surface-density v is the resolved part of the intensity of
magnetization 7 in the direction of the normal to the surface drawn
outwards, and the volume-density p is the ' convergence' (see
Art. 25) of the magnetization at a given point in the magnet.
This method of representing the action of a magnet as due
to a distribution of f magnetic matter ' is very convenient, but we
must always remember that it is only an artificial method of
representing the action of a system of polarized particles.
On the Action of one Magnetic Molecule o
387.] If, as in the chapter on Spherical Harmonics, Art. 129,
we make d , d d d
~TL = ^ T~ + m ~j — \- n r "> W
dh dx dy dz
where I, m, n are the direction-cosines of the axis It, then the
potential due to a magnetic molecule at the origin, whose axis is
parallel to klt and whose magnetic moment is mlt is
y _ d ml ml (
'** ~5*77~"HAi'
where A.L is the cosine of the angle between h± and r.
Again, if a second magnetic molecule whose moment is m2, and
whose axis is parallel to hz, is placed at the extremity of the radius
vector r, the potential energy due to the action of the one magnet
on the other is
387.] FORCE BETWEEN TWO MAGNETIZED PARTICLES. 11
(3)
(4)
where /u12 is the cosine of the angle which the axes make with each
other, and Xls A2 are the cosines of the angles which they make
with r.
Let us next determine the moment of the couple with which the
first magnet tends to turn the second round its centre.
Let us suppose the second magnet turned through an angle
d(f) in a plane perpendicular to a third axis &3, then the work done
against the magnetic forces will be -^ — dti, and the moment of the
a(f>
forces on the magnet in this plane will be
dW ml m2 ,dyl2 d\2^
~~d^ = ~^~\d$~ Al3^'
The actual moment acting on the second magnet may therefore
be considered as the resultant of two couples, of which the first
acts in a plane parallel to the axes of both magnets, and tends to
increase the angle between them with a force whose moment is
while the second couple acts in the plane passing through r and
the axis of the second magnet, and tends to diminish the angle
between these directions with a force
3 m* m9
>~^cos(r/h)siu(r/^, (7)
where (f^), (?'^2); (^1^2) denote the angles between the lines r,
To determine the force acting on the second magnet in a direction
parallel to a line 7/3, we have to calculate
dW d* ,K
(9)
(10)
If we suppose the actual force compounded of three forces, R,
H^ and H2, in the directions of r, ^ and ^2 respectively, then the
force in the direction of ^3 is
(11)
12 ELEMENTARY THEORY OF MAGNETISM. [388.
Since the direction of h% is arbitrary, we must have
3 tYl-i tlfli\ ~\
_/L ^^ .— — vMl2 "~~ 1 2/5
(12)
The force 72 is a repulsion, tending to increase r ; H^ and ZT2
act on the second magnet in the directions of the axes of the first
and second magnet respectively.
This analysis of the forces acting between two small magnets
was first given in terms of the Quaternion Analysis by Professor
Tait in the Quarterly Math. Journ. for Jan. 1860. See also his
work on Quaternions, Art. 414.
Particular Positions.
388.] (1) If Aj and A2 are each equal to 1, that is, if the axes
of the magnets are in one straight line and in the same direction,
fj.12 = 1, and the force between the magnets is a repulsion
p. TT , TT Qm1m2 . .
Jic-f jczi-f/ZgTs -- 4 -- (13)
The negative sign indicates that the force is an attraction.
(2) If A: and A2 are zero, and /*12 unity, the axes of the magnets
are parallel to each other and perpendicular to /, and the force
is a repulsion 3m1m2
In neither of these cases is there any couple.
(3) If A! = 1 and A2 = 0, then /u12 = 0. (15)
The force on the second magnet will be - — *— 2 in the direction
of its axis, and the couple will be — ^— 2 t tending to turn it parallel
to the first magnet. This is equivalent to a single force - ^ 2
acting parallel to the direction of the axis of the second magnet,
and cutting r at a point two-thirds of its length from m2.
Fig. 1.
Thus in the figure (1) two magnets are made to float on water,
388.]
FORCE BETWEEN TWO SMALL MAGNETS.
13
being in the direction of the axis of m1 , but having- its own axis
at right angles to that of ml. If two points, A, B, rigidly connected
with % and m2 respectively, are connected by means of a string T,
the system will be in equilibrium,, provided T cuts the line m1m2
at right angles at a point one-third of the distance from ml to m2 .
(4) If we allow the second magnet to turn freely about its centre
till it comes to a position of stable equilibrium, ?Fwill then be a
minimum as regards k2 , and therefore the resolved part of the force
due to m2, taken in the direction of ^15 will be a maximum. Hence,
if we wish to produce the greatest possible magnetic force at a
given point in a given direction by means of magnets, the positions
of whose centres are given, then, in order to determine the proper
directions of the axes of these magnets to produce this effect, we
have only to place a magnet in the given direction at the given
point, and to observe the direction of stable equilibrium of the
axis of a second magnet when its centre is placed at each of the
other given points. The magnets must then be placed with their
axes in the directions indicated by that of the second magnet.
Of course, in performing this experi
ment we must take account of terrestrial
magnetism, if it exists.
Let the second magnet be in a posi
tion of stable equilibrium as regards its
direction, then since the couple acting
on it vanishes, the axis of the second
magnet must be in the same plane with
that of the first. Hence
(M2) = (V)+M2), (16)
and the couple being
Fig. 2.
m
(sin (h-^ /t>2) — 3 cos (h-^ r) sin (r h2)),
(17)
we find when this is zero
tan (^ r) = 2 tan (r 7*2) ,
(18)
or tan^Wg-B = 2 ta,nRm2ff2. (19)
When this position has been taken up by the second magnet the
dV
value of W becomes
where h2 is in the direction of the line of force due to ml at
14 ELEMENTARY THEORY OF MAGNETISM. [389.
Hence W
,-.V;
T ~1
* (20)
Hence the second magnet will tend to move towards places of
greater resultant force.
The force on the second magnet may be decomposed into a force
R, which in this case is always attractive towards the first magnet,
and a force ffl parallel to the axis of the first magnet, where
H L = 3^ ** _ . (21)
^ 73 Ax2 + 1
In Fig. XVII, at the end of this volume, the lines of force and
equipotential surfaces in two dimensions are drawn. The magnets
which produce them are supposed to be two long cylindrical rods
the sections of which are represented by the circular blank spaces,
and these rods are magnetized transversely in the direction of the
arrows.
Jf we remember that there is a tension along the lines of force, it
is easy to see that each magnet will tend to turn in the direction
of the motion of the hands of a watch.
That on the right hand will also, as a whole, tend to move
towards the top, and that on the left hand towards the bottom
of the page.
On the Potential Energy of a Magnet placed in a Magnetic Field.
389.] Let V be the magnetic potential due to any system of
magnets acting on the magnet under consideration. We shall call
V the potential of the external magnetic force.
If a small magnet whose strength is m, and whose length is ds,
be placed so that its positive pole is at a point where the potential
is T3 and its negative pole at a point where the potential is F', the
potential energy of this magnet will be mCF—P'), or, if ds is
measured from the negative pole to the positive,
dV - ,1X
m-f-ds. (1)
as
If / is the intensity of the magnetization, and A, p, v its direc
tion-cosines, we may write,
mds =
dV dV dV dV
and -7- = A-y--f-ju-^ — |- v^->
ds dx dy dz
and, finally, if A, B, C are the components of magnetization,
A=\I, B=pl, C=vl,
390.] POTENTIAL ENERGY OP A MAGNET. 15
so that the expression (1) for the potential energy of the element
of the magnet becomes
To obtain the potential energy of a magnet of finite size, we
must integrate this expression for every element of the magnet.
We thus obtain
W = fff(A df + Bll^ + Cd-f) dxdydz (3)
J J J ^ dx dy dz '
as the value of the potential energy of the magnet with respect
to the magnetic field in which it is placed.
The potential energy is here expressed in terms of the components
of magnetization and of those of the magnetic force arising from
external causes.
By integration by parts we may express it in terms of the
distribution of magnetic matter and of magnetic potential
~ + -- + -dxdydzy (4)
where /, m, n are the direction-cosines of the normal at the element
of surface dS. If we substitute in this equation the expressions for
the surface- and volume-density of magnetic matter as given in
Art. 386, the expression becomes
pdS. (5)
We may write equation (3) in the form
+ Cy}dxdydz, (6)
where a, ft and y are the components of the external magnetic force.
On the Magnetic Moment and Axis of a Magnet.
390.] If throughout the whole space occupied by the magnet
the external magnetic force is uniform in direction and magnitude,
the components a, /3, y will be constant quantities, and if we write
IJJAdxdydz=lK, jjJBdxdydz=mK, [((cdxdydz = nKt (7)
the integrations being extended over the whole substance of the
magnet, the value of ^may be written
y). (8)
16 ELEMENTAEY THEORY OF MAGNETISM.
In this expression I, m, n are the direction-cosines of the axis of
the magnet, and K is the magnetic moment of the magnet. If
e is the angle which the axis of the magnet makes with the
direction of the magnetic force «£), the value of W may be written
JF = -K$cos€. (9)
If the magnet is suspended so as to be free to turn about a
vertical axis, as in the case of an ordinary compass needle, let
the azimuth of the axis of the magnet be $, and let it be inclined
0 to the horizontal plane. Let the force of terrestrial magnetism
be in a direction whose azimuth is 5 and dip £, then
a = «$p cos £ cos bj (3 = «£j cos £ sin 8, y = «£) sin f; (10)
I = cos 0 cos <£, m = cos 0 sin <£, n — sin 0 ; (11)
whence W— — KQ (cos £ cos 6 cos ($ — 8) + sin ( sin e). (12)
The moment of the force tending to increase $ by turning the
magnet round a vertical axis is
_ ^L=_K cos Ccos<9 sin (<J>-5). (13)
On the Expansion of the Potential of a Magnet in Solid Harmonics.
391.] Let V be the potential due to a unit pole placed at the
point (£, T?, f). The value of F" at the point #, y, z is
r= {(f-*)2+(>/-,?o2 +(<r-*)Ti (i)
This expression may be expanded in terms of spherical harmonics,
with their centre at the origin. We have then
(2)
when FQ = - , r being the distance of (f, 77, f ) from the origin, (3)
(4)
_
2~ 2r5
fee.
To determine the value of the potential energy when the magnet
is placed in the field of force expressed by this potential, we have
to integrate the expression for W in equation (3) with respect to
x, y and z, considering £, 77, (" and r as constants.
If we consider only the terms introduced by F~0, Ft and V2 the
result will depend on the following volume-integrals,
392.] EXPANSION OF THE POTENTIAL DUE TO A MAGNET. 17
lK = jjJAdxdydz, mK = fjfsdxdydz, nK =JJJ Cdxdydz; (6)
L=jjJAxdxdydz> M = jjj Bydxdydz, N =jjJCzdxdydz', (7)
P = (B* + Cy)dxdydz, Q =
R = ^y + Bnyndydz- (8)
We thus find for the value of the potential energy of the magnet
placed in presence of the unit pole at the point (^17, Q,
_
r5
This expression may also be regarded as the potential energy of
the unit pole in presence of the magnet, or more simply as the
potential at the point £ , 17, f due to the magnet.
On ike Centre of a Magnet and its Primary and Secondary Axes.
392.] This expression may be simplified by altering the directions
of the coordinates and the position of the origin. In the first
place, we shall make the direction of the axis of x parallel to the
axis of the magnet. This is equivalent to making
l—\^ m = 0, n — 0. (10)
If we change the origin of coordinates to the point (#', y', /), the
directions of the axes remaining unchanged, the volume-integrals
IK, mK and nK will remain unchanged, but the others will be
altered as follows :
L'=L-lKx', M'=M-mKy', Nf = N-nKz/-f (11)
P'=P—K(mz'+ny), Q'=Q- K(nx' + lz'\ R' — R— K(ly' + mx'}.
If we now make the direction of the axis of x parallel to the
axis of the magnet, and put
, Zl-M-N , R , Q , .
x'= —^ > y = Tr> z = -^> (13)
2A A A
then for the new axes M and N have their values unchanged, and
the value of 1! becomes \ (M+N). P remains unchanged, and Q
and R vanish. We may therefore write the potential thus,
VOL. II.
18 ELEMENTARY THEOEY OF MAGNETISM. \_392-
We have thus found a point, fixed with respect to the magnet,
such that the second term of the potential assumes the most simple
form when this point is taken as origin of coordinates. This point
we therefore define as the centre of the magnet, and the axis
drawn through it in the direction formerly defined as the direction
of the magnetic axis may be defined as the principal axis of the
magnet.
We may simplify the result still more by turning the axes of y
and z round that of x through half the angle whose tangent is
p
-=£ — — . This will cause P to become zero, and the final form
of the potential may be written
Kt ttf-
3 2
This is the simplest form of the first two terms of the potential
of a magnet. When the axes of y and z are thus placed they may
be called the Secondary axes of the magnet.
We may also determine the centre of a magnet by finding the
position of the origin of coordinates, for which the surface-integral
of the square of the second term of the potential, extended over
a sphere of unit radius, is a minimum.
The quantity which is to be made a minimum is, by Art. 141,
4 (Z2 + Mz + N*-MN-NL-LM] + 3 (P2 + Q2 +^2). (16)
The changes in the values of this quantity due to a change of
position of the origin may be deduced from equations (11) and (12).
Hence the conditions of a minimum are
21(2 L—M— N)+3nQ+3mR = 0,
2m(2M-N-L)+3lR+3nP = 0, (17)
2n (2N—Z—M)+3mP+3lQ = 0.
If we assume I = I, m = 0, n = Q, these conditions become
2L-M—N=0, q = 0, R=0, (18)
which are the conditions made use of in the previous invest
igation.
This investigation may be compared with that by which the
potential of a system of gravitating matter is expanded. In the
latter case, the most convenient point to assume as the origin
is the centre of gravity of the system, and the most convenient
axes are the principal axes of inertia through that point.
In the case of the magnet, the point corresponding to the centre
of gravity is at an infinite distance in the direction of the axis,
394'] CONVENTION RESPECTING SIGNS. 19
and the point which we call the centre of the magnet is a point
having- different properties from those of the centre of gravity.
The quantities If, M, N correspond to the moments of inertia,
and P, Q, R to the products of inertia of a material body, except
that Z, M and N are not necessarily positive quantities.
When the centre of the magnet is taken as the origin, the
spherical harmonic of the second order is of the sectorial form,
having its axis coinciding with that of the magnet, and this is
true of no other point.
When the magnet is symmetrical on all sides of this axis, as
in the case of a figure of revolution, the term involving the harmonic
of the second order disappears entirely.
393.] At all parts of the earth's surface, except some parts of
the Polar regions, one end of a magnet points towards the north,
or at least in a northerly direction, and the other in a southerly
direction. In speaking of the ends of a magnet we shall adopt the
popular method of calling the end which points to the north the
north end of the magnet. When, however, we speak in the
language of the theory of magnetic fluids we shall use the words
Boreal and Austral. Boreal magnetism is an imaginary kind of
matter supposed to be most abundant in the northern, parts of
the earth, and Austral magnetism is the imaginary magnetic
matter which prevails in the southern regions of the earth. The
magnetism of the north end of a magnet is Austral, and that of
the south end is Boreal. When therefore we speak of the north
and south ends of a magnet we do not compare the magnet with
the earth as the great magnet, but merely express the position
which the magnet endeavours to take up when free to move. When,
on the other hand, we wish to compare the distribution of ima
ginary magnetic fluid in the magnet with that in the earth we shall
use the more grandiloquent words Boreal and Austral magnetism.
394.] In speaking of a field of magnetic force we shall use the
phrase Magnetic North to indicate the direction in which the
north end of a compass needle would point if placed in the field
of force.
In speaking of a line of magnetic force we shall always suppose
it to be traced from magnetic south to magnetic north, and shall
call this direction positive. In the same way the direction of
magnetization of a magnet is indicated by a line drawn from the
south end of the magnet towards the north end, and the end of
the magnet which points north is reckoned the positive end.
20 ELEMENTARY THEORY OF MAGNETISM. \_394--
We shall consider Austral magnetism, that is, the magnetism of
that end of a magnet which points north, as positive. If we denote
its numerical value by m> then the magnetic potential
and the positive direction of a line of force is that in which V
diminishes.
CHAPTER II.
MAGNETIC FORCE AND MAGNETIC INDUCTION.
395.] WE have already (Art. 386) determined the magnetic
potential at a given point due to a magnet, the magnetization of
which is given at every point of its substance, and we have shewn
that the mathematical result may be expressed either in terms
of the actual magnetization of every element of the magnet, or
in terms of an imaginary distribution of ' magnetic matter,' partly
condensed on the surface of the magnet and partly diffused through
out its substance.
The magnetic potential, as thus denned, is found by the same
mathematical process, whether the given point is outside the magnet
or within it. The force exerted on a unit magnetic pole placed
at any point outside the magnet is deduced from the potential by
the same process of differentiation as in the corresponding electrical
problem. If the components of this force are a, /3, y,
dV dV dV m
a= — > /3 = j-j y— j-- (1)
dx dy dz
To determine by experiment the magnetic force at a point within
the magnet we must begin by removing part of the magnetized
substance, so as to form a cavity within which we are to place the
magnetic pole. The force acting on the pole will depend, in general,
in the form of this cavity, and on the inclination of the walls of
the cavity to the direction of magnetization. Hence it is necessary,
in order to avoid ambiguity in speaking of the magnetic force
within a magnet, to specify the form and position of the cavity
within which the force is to be measured. It is manifest that
when the form and position of the cavity is specified, the point
within it at which the magnetic pole is placed must be regarded as
22 MAGNETIC FORCE AND MAGNETIC INDUCTION. [396.
no longer within the substance of the magnet, and therefore the
ordinary methods of determining the force become at once applicable.
396.] Let us now consider a portion of a magnet in which the
direction and intensity of the magnetization are uniform. Within
this portion let a cavity be hollowed out in the form of a cylinder,
the axis of which is parallel to the direction of magnetization, and
let a magnetic pole of unit strength be placed at the middle point
of the axis.
Since the generating lines of this cylinder are in the direction
of magnetization, there will be no superficial distribution of mag
netism on the curved surface, and since the circular ends of the
cylinder are perpendicular to the direction of magnetization, there
will be a uniform superficial distribution, of which the surface-
density is /for the negative end, and —/for the positive end.
Let the length of the axis of the cylinder be 2 b, and its radius a.
Then the force arising from this superficial distribution on a
magnetic pole placed at the middle point of the axis is that due
to the attraction of the disk on the positive side, and the repulsion
of the disk on the negative side. These two forces are equal and
in the same direction, and their sum is
---!=. (2)
From this expression it appears that the force depends, not on
the absolute dimensions of the cavity, but on the ratio of the length
to the diameter of the cylinder. Hence, however small we make the
cavity, the force arising from the surface distribution on its walls
will remain, in general, finite.
397.] We have hitherto supposed the magnetization to be uniform
and in the same direction throughout the whole of the portion of
the magnet from which the cylinder is hollowed out. Wlien the
magnetization is not thus restricted, there will in general be a
distribution of imaginary magnetic matter through the substance
of the magnet. The cutting out of the cylinder will remove part
of this distribution, but since in similar solid figures the forces at
corresponding points are proportional to the linear dimensions of
the figures, the alteration of the force on the magnetic pole due
to the volume-density of magnetic matter will diminish indefinitely
as the size of the cavity is diminished, while the effect due to
the surface-density on the walls of the cavity remains, in general,
finite.
If, therefore, we assume the dimensions of the cylinder so small
399-1 MAGNETIC FORCE IN A CAVITY. 23
that the magnetization of the part removed may be regarded as
everywhere parallel to the axis of the cylinder, and of constant
magnitude I, the force on a magnetic pole placed at the middle
point of the axis of the cylindrical hollow will be compounded
of two forces. The first of these is that due to the distribution
of magnetic matter on the outer surface of the magnet, and
throughout its interior, exclusive of the portion hollowed out. The
components of this force are a, /3 and y, derived from the potential
by equations (1). The second is the force 72, acting along the axis
of the cylinder in the direction of magnetization. The value of
this force depends on the ratio of the length to the diameter of the
cylindric cavity.
398.] Case I. Let this ratio be very great, or let the diameter
of the cylinder be small compared with its length. Expanding the
expression for R in terms of j- , it becomes
a quantity which vanishes when the ratio of b to a is made infinite.
Hence, when the cavity is a very narrow cylinder with its axis parallel
to the direction of magnetization, the magnetic force within the
cavity is not affected by the surface distribution on the ends of the
cylinder, and the components of this force are simply a, /3, y, where
dV dV dV ,,.
a = -- 7-, 0 = — -=-, y= — -—. (4)
dx dy dz
We shall define the force within a cavity of this form as the
magnetic force within the magnet. Sir William Thomson has
called this the Polar definition of magnetic force. When we have
occasion to consider this force as a vector we shall denote it
*>7$.
399.] Case II. Let the length of the cylinder be very small
compared with its diameter, so that the cylinder becomes a thin
disk. Expanding the expression for R in terms of - , it becomes
_ £+££-*..}, (5)
a 2 #3 3
the ultimate value of which, when the ratio of a to b is made
infinite, is 4 TT J.
Hence, when the cavity is in the form of a thin disk, whose plane
is normal to the direction of magnetization, a unit magnetic pole
24 MAGNETIC FORCE AND MAGNETIC INDUCTION. [400.
placed at the middle of the axis experiences a force 4 IT I in the
direction of magnetization arising from the superficial magnetism
on the circular surfaces of the disk *.
Since the components of J are A, B and (7, the components of
this force are 4 -n A, 4 TT B and 4 TT C. This must be compounded
with the force whose components are a, {3, y.
400.] Let the actual force on the unit pole be denoted by the
vector 35, and its components by a, b and c, then
a = a + 4 TT A,
0=/3 + 47T.£, (6)
C = y -f 4 TT C.
We shall define the force within a hollow disk, whose plane sides
are normal to the direction of magnetization, as the Magnetic
Induction within the magnet. Sir William Thomson has called
this the Electromagnetic definition of magnetic force.
The three vectors, the magnetization 3, the magnetic force <!fj,
and the magnetic induction S3 are connected by the vector equation
47:3. (7)
Line-Integral of Magnetic Force.
401.] Since the magnetic force, as denned in Art. 398, is that
due to the distribution of free magnetism on the surface and through
the interior of the magnet, and is not affected by the surface-
magnetism of the cavity, it may be derived directly from the
general expression for the potential of the magnet, and the line-
integral of the magnetic force taken along any curve from the
point A to the point B is
where VA and V^ denote the potentials at A and B respectively.
* On the force within cavities of other forms.
1. Any narrow crevasse. The force arising from the surface-magnetism is
47r/cos€ in the direction of the normal to the plane of the crevasse, where 6 is the
angle between this normal and the direction of magnetization. When the crevasse
is parallel to the direction of magnetization the force is the magnetic force £ ; when
the crevasse is perpendicular to the direction of magnetization the force is the
magnetic induction 93.
2. In an elongated cylinder, the axis of which makes an angle « with the
direction of magnetization, the force arising from the surface-magnetism is 27r/sin e,
perpendicular to the axis in the plane containing the axis and the direction of
magnetization.
3. In a sphere the force arising from surface-magnetism is f IT I in the direction of
magnetization.
402.] SURF ACE -INTEGRAL. 25
Surface-Integral of Magnetic Induction.
402.] The magnetic induction through the surface 8 is defined
as the value of the integral
Q = ff%cos€dS, (9)
where 23 denotes the magnitude of the magnetic induction at the
element of surface clS, and e the angle between the direction of
the induction and the normal to the element of surface, and the
integration is to be extended over the whole surface, which may
be either closed or bounded by a closed curve.
If a, b, c denote the components of the magnetic induction, and
/, m, n the direction-cosines of the normal, the surface-integral
may be written
q = jj(la+mb+nG)d8. (10)
If we substitute for the components of the magnetic induction
their values in terms of those of the magnetic force, and the
magnetization as given in Art. 400, we find
Q = n(la + mp + ny)dS + 4 TT (lA + m£ + nC)dS. (11)
We shall now suppose that the surface over which the integration
extends is a closed one, and we shall investigate the value of the
two terms on the right-hand side of this equation.
Since the mathematical form of the relation between magnetic
force and free magnetism is the same as that between electric
force and free electricity, we may apply the result given in Art. 77
to the first term in the value of Q by substituting a, ft, y, the
components of magnetic force, for X, Y, Z, the components of
electric force in Art. 77, and M, the algebraic sum of the free
magnetism within the closed surface, for e, the algebraic sum of
the free electricity.
We thus obtain the equation
ny)48*x 4irM. (12)
Since every magnetic particle has two poles, which are equal
in numerical magnitude but of opposite signs, the algebraic sum
of the magnetism of the particle is zero. Hence, those particles
which are entirely within the closed surface S can contribute
nothing to the algebraic sum of the magnetism within S. The
26 MAGNETIC FORCE AND MAGNETIC INDUCTION. [403.
value of M must therefore depend only on those magnetic particles
which are cut by the surface S.
Consider a small element of the magnet of length s and trans
verse section kz, magnetized in the direction of its length, so that
the strength of its poles is m. The moment of this small magnet
will be ms, and the intensity of its magnetization, being the ratio
of the magnetic moment to the volume, will be
/=£• (13)
Let this small magnet be cut by the surface S, so that the
direction of magnetization makes an angle e' with the normal
drawn outwards from the surface, then if dS denotes the area of
the section, p = ds cos e/t ( 1 4)
The negative pole — m of this magnet lies within the surface S.
Hence, if we denote by dM the part of the free magnetism
within S whic*h is contributed by this little magnet,
IS. (15)
To find M, the algebraic sum of the free magnetism within the
closed surface S, we must integrate this expression over the closed
surface, so that
M=-
or writing A, .Z?, C for the components of magnetization, and I, m, n
for the direction-cosines of the normal drawn outwards,
(16)
This gives us the value of the integral in the second term of
equation (11). The value of Q in that equation may therefore
be found in terms of equations (12) and (16),
Q = 47r3/-47rl/= 0, (17)
or, the surface-integral of the magnetic induction through any closed
surface is zero.
403.] If we assume as the closed surface that of the differential
element of volume dx dy dz, we obtain the equation
*! + *+* = 0. (18)
dx dy dz
This is the solenoidal condition which is always satisfied by the
components of the magnetic induction.
405.] LINES OF MAGNETIC INDUCTION. 27
Since the distribution of magnetic induction is solenoidal, the
induction through any surface bounded by a closed curve depends
only on the form and position of the closed curve, and not on that
of the surface itself.
404.] Surfaces at every point of which
la + mb + nc = 0 (19)
are called Surfaces of no induction, and the intersection of two such
surfaces is called a Line of induction. The conditions that a curve,
Sj may be a line of induction are
1 dx 1 dy \ dz , .
= 'L = . (20)
a ds I ds c ds
A system of lines of induction drawn through every point of a
closed curve forms a tubular surface called a Tube of induction.
The induction across any section of such a tube is the same.
If the induction is unity the tube is called a Unit tube of in
duction.
All that Faraday * says about lines of magnetic force and mag
netic sphondyloids is mathematically true, if understood of the
lines and tubes of magnetic induction.
The magnetic force and the magnetic induction are identical
outside the magnet, but within the substance of the magnet they
must be carefully distinguished. In a straight uniformly mag
netized bar the magnetic force due to the magnet itself is from
the end which points north, which we call the positive pole, towards
the south end or negative pole, both within the magnet and in
the space without.
The magnetic induction, on the other hand, is from the positive
pole to the negative outside the magnet, and from the negative
pole to the positive within the magnet, so that the lines and tubes
of induction are re-entering or cyclic figures.
The importance of the magnetic induction as a physical quantity
will be more clearly seen when we study electromagnetic phe
nomena. When the magnetic field is explored by a moving wire,
as in Faraday's Exp. Res. 3076, it is the magnetic induction and
not the magnetic force which is directly measured.
The Vector-Potential of Magnetic Induction.
405.] Since, as we have shewn in Art. 403, the magnetic in
duction through a surface bounded by a closed curve depends on
* Exp. Res., series xxviii.
28 MAGNETIC FORCE AND MAGNETIC INDUCTION. [406.
the closed curve, and not on the form of the surface which is
bounded by it, it must be possible to determine the induction
through a closed curve by a process depending only on the nature
of that curve, and not involving the construction of a surface
forming a diaphragm of the curve.
This may be done by finding a vector 21 related to 33, the magnetic
induction, in such a way that the line-integral of SI, extended round
the closed curve, is equal to the surface-integral of 33, extended
over a surface bounded by the closed curve.
If, in Art. 24, we write F9 G, H for the components of SI, and
a, b, c for the components of 33, we find for the relation between
these components
dH dG dF dH dG dF
a= —
.j 7
dz dz ax ax ay
The vector SI, whose components are F, G, //, is called the vector-
potential of magnetic induction. The vector-potential at a given
point, due to a magnetized particle placed at the origin, is nume
rically equal to the magnetic moment of the particle divided by
the square of the radius vector and multiplied by the sine of the
angle between the axis of magnetization and the radius vector,
and the direction of the vector-potential is perpendicular to the
plane of the axis of magnetization and the radius vector, and is
such that to an eye looking in the positive direction along the
axis of magnetization the vector-potential is drawn in the direction
of rotation of the hands of a watch.
Hence, for a magnet of any form in which A^ B, C are the
components of magnetization at the point xyz, the components
of the vector-potential at the point f 77 £ are
(22)
where p is put, for conciseness, for the reciprocal of the distance
between the points (f, 77, Q and (#, y, z), and the integrations are
extended over the space occupied by the magnet.
406.] The scalar, or ordinary, potential of magnetic force,
Art. 386, becomes when expressed in the same notation,
406.] VECTOR- POTENTIAL. 29
/v /y\ t-j /v\
Kemembering that ~ = — -~, and that the integral
dx u/ £
has the value — 4 TT ( A) when the point (£, 77, f) is included within
the limits of integration, and is zero when it is not so included,
(A) being the value of A at the point (f, 77, (*), we find for the value
of the ^-component of the magnetic induction,
dH _ dG_
dr] d£
f d^p dzp \ d'*p d2j) }
\dydr) dzdC' dx dr] dxd^S
7> r, ^ 7 7
-ri - ~ + B -/- -f- (7 7 \dxdydz
d£jJJ ( dx dy d
The first term of this expression is evidently -- ^ , or a, the
component of the magnetic force.
The quantity under the integral sign in the second term is zero
for every element of volume except that in which the point (f, ry, £)
is included. If the value of A at the point (f, r/, f) is (A), the
value of the second term is 4 TT (A)9 where (A) is evidently zero
at all points outside the magnet.
We may now write the value of the ^-component of the magnetic
induction « = o+4w(^), (25)
an equation which is identical with the first of those given in
Art. 400. The equations for b and c will also agree with those
of Art. 400.
We have already seen that the magnetic force § is derived from
the scalar magnetic potential V by the application of Hamilton's
operator y , so that we may write, as in Art. 1 7,
£=-vF, (26)
and that this equation is true both without and within the magnet.
It appears from the present investigation that the magnetic
induction S3 is derived from the vector-potential SI by the appli
cation of the same operator, and that the result is true within the
magnet as well as without it.
The application of this operator to a vector-function produces,
30 MAGNETIC FORCE AND MAGNETIC INDUCTION. [406.
in general, a scalar quantity as well as a vector. The scalar part,
however, which we have called the convergence of the vector-
function, vanishes when the vector-function satisfies the solenoidal
condition
dF dG dH
•Jl + -J~ + -7TF = °*
df; dr] d£
By differentiating the expressions for F, G, If in equations (22), we
find that this equation is satisfied by these quantities.
We may therefore write the relation between the magnetic
induction and its vector-potential
23 = V %
which may be expressed in words by saying that the magnetic
induction is the curl of its vector-potential. See Art. 25.
CHAPTER III
MAGNETIC SOLENOIDS AND SHELLS*.
On Particular Forms of Magnets.
407.] IF a long narrow filament of magnetic matter like a wire
is magnetized everywhere in a longitudinal direction, then the
product of any transverse section of the filament into the mean
intensity of the magnetization across it is called the strength of
the magnet at that section. If the filament were cut in two at
the section without altering the magnetization, the two surfaces,
when separated, would be found to have equal and opposite quan
tities of superficial magnetization, each of which is numerically
equal to the strength of the magnet at the section.
A filament of magnetic matter, so magnetized that its strength
is the same at every section, at whatever part of its length the
section be made, is called a Magnetic Solenoid.
If m is the strength of the solenoid, ds an element of its length,
r the distance of that element from a given point, and e the angle
which r makes with the axis of magnetization of the element, the
potential at the given point due to the element is
m ds cos € m dr ..
— o = —s- ~r~ ds.
r2 r* ds
Integrating this expression with respect to s} so as to take into
account all the elements of the solenoid, the potential is found
to be ,11^
V = m ( ) >
rl r2
T! being the distance of the positive end of the solenoid, and r^
that of the negative end from the point where V exists.
Hence the potential due to a solenoid, and consequently all its
magnetic effects, depend only on its strength and the position of
* See Sir W. Thomson's 'Mathematical Theory of Magnetism,' Phil. Trans., 1850,
or Reprint.
32 MAGNETIC SOLENOIDS AND SHELLS. [408.
its ends, and not at all on its form, whether straight or curved,
between these points.
Hence the ends of a solenoid may be called in a strict sense
its poles.
If a solenoid forms a closed curve the potential due to it is zero
at every point, so that such a solenoid can exert no magnetic
action, nor can its magnetization be discovered without breaking
it at some point and separating the ends.
If a magnet can be divided into solenoids, all of which either
form closed curves or have their extremities in the outer surface
of the magnet, the magnetization is said to be solenoidal, and,
since the action of the magnet depends entirely upon that of the
ends of the solenoids, the distribution of imaginary magnetic matter
will be entirely superficial.
Hence the condition of the magnetization being solenoidal is
dA dB dC _
dx dy dz
where A, B, C are the components of the magnetization at any
point of the magnet.
408.] A longitudinally magnetized filament, of which the strength
varies at different parts of its length, may be conceived to be made
up of a bundle of solenoids of different lengths, the sum of the
strengths of all the solenoids which pass through a given section
being the magnetic strength of the filament at that section. Hence
any longitudinally magnetized filament may be called a Complex
Solenoid.
If the strength of a complex solenoid at any section is m, then
the potential due to its action is
ds where m is variable,
Cm dr
f% -
m\ mi /I
fll* 4* i 4*
/I /*> J I
l dm 7
ds
This shews that besides the action of the two ends, which may
in this case be of different strengths, there is an action due to the
distribution of imaginary magnetic matter along the filament with
a linear density dm
/V. — - •"— j *
ds
Magnetic Shells.
409.] If a thin shell of magnetic matter is magnetized in a
SHELLS. 33
direction everywhere normal to its surface, the intensity of the
magnetization at any place multiplied by the thickness of the
sheet at that place is called the Strength of the magnetic shell
at that place.
If the strength of a shell is everywhere equal, it is called a
Simple magnetic shell; if it varies from point to point it may be
conceived to be made up of a number of simple shells superposed
and overlapping each other. It is therefore called a Complex
magnetic shell.
Let dS be an element of the surface of the shell at Q, and 4>
the strength of the shell, then the potential at any point, P, due
to the element of the shell, is
d V = <J> — - dS cos €*
r2
where e is the angle between the vector QP, or r and the normal
drawn from the positive side of the shell.
But if du> is the solid angle subtended by dS at the point P
r2 da — dS cos e,
whence dF = <&da>,
and therefore in the case of a simple magnetic shell
or, the potential due to a magnetic shell at any point is the product
of its strength into the solid angle subtended by its edge at the
given point*.
410.] The same result may be obtained in a different way by
supposing the magnetic shell placed in any field of magnetic force,
and determining the potential energy due to the position of the
shell.
If V is the potential at the element dS, then the energy due to
this element is dy dy dy
* (^ -r- +m~j- + n ~r) <***
\ da dy dz'
or, the product of the strength of the shell into the part of the
surface-integral of V due to the element dS of the shell.
Hence, integrating with respect to all such elements, the energy
due to the position of the shell in the field is equal to the product
of the strength of the shell and the surf ace -integral of the magnetic
induction taken over the surface of the shell.
Since this surface-integral is the same for any two surfaces which
* This theorem is due to Gauss, General Theory of Terrestrial Magnetism, § 38.
VOL. II. D
34 MAGNETIC SOLENOIDS AND SHELLS. [411-
have the same bounding- edge and do not include between them
any centre of force, the action of the magnetic shell depends only
on the form of its edge.
Now suppose the field of force to be that due to a magnetic
pole of strength m. We have seen (Art. 76, Cor.) that the surface-
integral over a surface bounded by a given edge is the product
of the strength of the pole and the solid angle subtended by the
edge at the pole. Hence the energy due to the mutual action
of the pole and the shell is
and this (by Green's theorem. Art. 100) is equal to the product
of the strength of the pole into the potential due to the shell at
the pole. The potential due to the shell is therefore 4> co.
411.] If a magnetic pole m starts from a point on the negative
surface of a magnetic shell, and travels along any path in space so as
to come round the edge to a point close to where it started but on
the positive side of the shell, the solid angle will vary continuously,
and will increase by 4 TT during the process. The work done by
the pole will be 4 TT 4> m, and the potential at any point on the
positive side of the shell will exceed that at the neighbouring point
on the negative side by 4 TT 4>.
If a magnetic shell forms a closed surface, the potential outside
the shell is everywhere zero, and that in the space within is
everywhere 4 TT 4>, being positive when the positive side of the shell
is inward. Hence such a shell exerts no action on any magnet
placed either outside or inside the shell.
412.] If a magnet can be divided into simple magnetic shells,
either closed or having their edges on the surface of the magnet,
the distribution of magnetism is called Lamellar. If <£ is the
sum of the strengths of all the shells traversed by a point in
passing from a given point to a point xy z by a line drawn within
the magnet, then the conditions of lamellar magnetization are
,_<Z<I> d<}> d(f>
A = — =— , JD = -r— , L> = — T~ *
dx dy dz
The quantity, <J>, which thus completely determines the magnet
ization at any point may be called the Potential of Magnetization.
It must be carefully distinguished from the Magnetic Potential.
413.] A magnet which can be divided into complex magnetic
shells is said to have a complex lamellar distribution of mag
netism. The condition of such a distribution is that the lines of
415.] POTENTIAL DUE TO A LAMELLAE MAGNET. 35
magnetization must be such that a system of surfaces can be drawn
cutting them at right angles. This condition is expressed by the
well-known equation
Aff__<lB} ^A_<IC ^_<U
^dy dz> ^dz dx' ^dx dy '
Forms of the Potentials of Solenoidal and Lamellar Magnets.
414.] The general expression for the scalar potential of a magnet
where p denotes the potential at (#, y, z) due to a unit magnetic
pole placed at f, TJ, £ or in other words, the reciprocal of the
distance between (f, r;, Q, the point at which the potential is
measured, and (#, y> z), the position of the element of the magnet
to which it is due.
This quantity may be integrated by parts, as in Arts. 96, 386.
where I, m, n are the direction-cosines of the normal drawn out
wards from dS, an element of the surface of the magnet.
When the magnet is solenoidal the expression under the integral
sign in the second term is zero for every point within the magnet,
so that the triple integral is zero, and the scalar potential at any
point, whether outside or inside the magnet, is given by the surface-
integral in the first term.
The scalar potential of a solenoidal magnet is therefore com
pletely determined when the normal component of the magnet
ization at every point of the surface is known, and it is independent
of the form of the solenoids within the magnet.
415.] In the case of a lamellar magnet the magnetization is
determined by c/>, the potential of magnetization, so that
dcf) d<j> d$
•**• -— ~^ — j .£> = —7— , <-/ = — ; — •
ax ay dz
The expression for V may therefore be written
= fff,
JJJ \
dp .
'
dx dx dy dy dz dz
Integrating this expression by parts, we find
D 2
36 MAGNETIC SOLENOIDS AND SHELLS.
The second term is zero unless the point (f, r/, f) is included in
the magnet, in which case it becomes 4 TT (<£) where (<£) is the value
of <p at the point £, 77, f The surface-integral may be expressed in
terms of rt the line drawn from (x, y, z] to (f, rj, f ), and 0 the angle
which this line makes with the normal drawn outwards from dSt
so that the potential may be written
where the second term is of course zero when the point (f, TJ, f) is
not included in the substance of the magnet.
The potential, F, expressed by this equation, is continuous even
at the surface of the magnet, where $ becomes suddenly zero, for
if we write
fit =
and if £1L is the value of H at a point just within the surface, and
122 that at a point close to the first but outside the surface,
fla = ^ + 477^),
r2 = r,.
The quantity H is not continuous at the surface of the magnet.
The components of magnetic induction are related to 12 by the
equations
d& da da
a= -- =— , 0= -- =-, c — -- -j- •
dx dy dz
416.] In the case of a lamellar distribution of magnetism we
may also simplify the vector-potential of magnetic induction.
Its ^-component may be written
By integration by parts we may put this in the form of the
surface-integral
or F .
The other components of the vector-potential may be written
down from these expressions by making the proper substitutions.
On Solid Angles.
417.] We have already proved that at any point P the potential
4 1 8.] SOLID ANGLES. 37
due to a magnetic shell is equal to the solid angle subtended by
the edge of the shell multiplied by the strength of the shell. As
we shall have occasion to refer to solid angles in the theory of
electric currents, we shall now explain how they may be measured.
Definition. The solid angle subtended at a given point by a
closed curve is measured by the area of a spherical surface whose
centre is the given point and whose radius is unity, the outline
of which is traced by the intersection of the radius vector with the
sphere as it traces the closed curve. This area is to be reckoned
positive or negative according as it lies on the left or the right-
hand of the path of the radius vector as seen from the given point.
Let (£, r], f) be the given point, and let (#, y, z) be a point on
the closed curve. The coordinates- x, y, z are functions of s, the
length of the curve reckoned from a given point. They are periodic
functions of s, recurring whenever s is increased by the whole length
of the closed curve.
We may calculate the solid angle o> directly from the definition
thus. Using spherical coordinates with centre at (£, 77, Q, and
putting
x — f = r sin0cos$, y— rj = r sin 0 sin^, z — C=rcos0,
we find the area of any curve on the sphere by integrating
co = /(I— cos0) d$,
or, using the rectangular coordinates,
the integration being extended round the curve s.
If the axis of z passes once through the closed curve the first
term is 2 IT. If the axis of z does not pass through it this term
is zero.
418.] This method of calculating a solid angle involves a choice
of axes which is to some extent arbitrary, and it does not depend
solely on the closed curve. Hence the following method, in which
no surface is supposed to be constructed, may be stated for the sake
of geometrical propriety.
As the radius vector from the given point traces out the closed
curve, let a plane passing through the given point roll on the
closed curve so as to be a tangent plane at each point of the curve
in succession. Let a line of unit-length be drawn from the given
point perpendicular to this plane. As the plane rolls round the
38 MAGNETIC SOLENOIDS AND SHELLS. [4 1 9.
closed curve the extremity of the perpendicular will trace a second
closed curve. Let the length of the second closed curve be o-, then
the solid angle subtended by the first closed curve is
00 = 27T — (7.
This follows from the well-known theorem that the area of a
closed curve on a sphere of unit radius, together with the circum
ference of the polar curve, is numerically equal to the circumference
of a great circle of the sphere.
This construction is sometimes convenient for calculating the
solid angle subtended by a rectilinear figure. For our own purpose,
which is to form clear ideas of physical phenomena, the following
method is to be preferred, as it employs no constructions which do
not flow from the physical data of the problem.
419.] A closed curve s is given in space, and we have to find
the solid angle subtended by s at a given point P.
If we consider the solid angle as the potential of a magnetic shell
of unit strength whose edge coincides with the closed curve, we
must define it as the work done by a unit magnetic pole against
the magnetic force while it moves from an infinite distance to the
point P. Hence, if cr is the path of the pole as it approaches the
point P, the potential must be the result of a line-integration along
this path. It must also be the result of a line-integration along
the closed curve s. The proper form of the expression for the solid
angle must therefore be that of a double integration with respect
to the two curves s and a.
When P is at an infinite distance, the solid angle is evidently
zero. As the point P approaches, the closed curve, as seen from
the moving point, appears to open out, and the whole solid angle
may be conceived to be generated by the apparent motion of the
different elements of the closed curve as the moving point ap
proaches.
As the point P moves from P to P' over the element do-, the
element QQ' of the closed curve, which we denote by ds, will
change its position relatively to P, and the line on the unit sphere
corresponding to QQ' will sweep over an area on the spherical
surface, which we may write
da = Udsdcr. (I)
To find FT let us suppose P fixed while the closed curve is moved
parallel to itself through a distance da- equal to PPf but in the
opposite direction. The relative motion of the point P will be the
same as in the real case.
420.]
GENERATION OF A SOLID ANGLE.
39
During this motion the element QQ' will generate an area in
the form of a parallelogram whose sides are parallel and equal
to Q Q' and PP'. If we construct a pyramid on this parallelogram
as base with its vertex at P, the solid angle of this pyramid will
be the increment d& which we are in search of.
To determine the value of this solid
angle, let 6 and tf be the angles which
ds and dcr make with PQ respect
ively, and let <£ be the angle between
the planes of these two angles, then
the area of the projection of the
parallelogram ds .dcr on a. plane per
pendicular to PQ or r will be
ds dcr sin Q sin 6' sin
and since this is equal to r2 d<a, we find
Fig. 3.
Hence
du> = II ds dcr = -g sin Q sin 6' sin </> ds dcr.
n = — - sin 6 sin 0' sin <>.
(2)
(3)
420.] We may express the angles 6, 6', and $ in terms of
and its differential coefficients with respect to s and o-, for
cos0= -=-,
/»/
cos<9'= •-=-,
dcr
and sin 6 sin 6' cos cp = r
dsdcr
(4)
We thus find the following value for D2,
(5)
A third expression for II in terms of rectangular coordinates
may be deduced from the consideration that the volume of the
pyramid whose solid angle is d& and whose axis is r is
J r* do) = J r* FT ds dcr.
But the volume of this pyramid may also be expressed in terms
of the projections of r, ds, and dcr on the axis of #, y and zt as
a determinant formed by these nine projections, of which we must
take the third part. We thus find as the value of n,
n = -^
-=— > -^— > -=—
c— *i
T\—y>
<* -.
l— *>
-7— >
dcr
drj
-j— >
dcr
T«'
dx
Ts*
d_y_
7 ^
ds
dz
~ds"
(6)
40 MAGNETIC SOLENOIDS AND SHELLS. [421.
This expression gives the value of FT free from the ambiguity of
sign introduced by equation (5).
421.] The value of o>, the solid angle subtended by the closed
curve at the point P, may now be written
a) = ndsdv-i-WQ, (7)
where the integration with respect to s is to be extended completely
round the closed curve, and that with respect to <r from A a fixed
point on the curve to the point P. The constant <o0 is the value
of the solid angle at the point A. It is zero if A is at an infinite
distance from the closed curve.
The value of o> at any point P is independent of the form of
the curve between A and P provided that it does not pass through
the magnetic shell itself. If the shell be supposed infinitely thin,
and if P and Pf are two points close together, but P on the positive
and P' on the negative surface of the shell, then the curves AP and
AP/ must lie on opposite sides of the edge of the shell, so that PAP'
is a line which with the infinitely short line PP forms a closed
circuit embracing the edge. The value of o> at P exceeds that at P'
by 47T, that is, by the surface of a sphere of radius unity.
Hence, if a closed curve be drawn so as to pass once through
the shell, or in other words, if it be linked once with the edge
of the shell, the value of the integral I lUdsdv extended round
both curves will be 47r.
This integral therefore, considered as depending only on the
closed curve s and the arbitrary curve AP, is an instance of a
_ function of multiple values, since, if we pass from A to P along
different paths the integral will have different values according
to the number of times which the curve AP is twined round the
curve s.
If one form of the curve between A and P can be transformed
into another by continuous motion without intersecting the curve
s, the integral will have the same value for both curves, but if
during the transformation it intersects the closed curve n times the
values of the integral will differ by 47m.
If s and a- are any two closed curves in space, then, if they are
not linked together, the integral extended once round both is
zero.
If they are intertwined n times in the same direction, the value
of the integral is 4iTn. It is possible, however, for two curves
422.] VECTOR- POTENTIAL OF A CLOSED CURVE. 41
to be intertwined alternately in opposite directions, so that they
are inseparably linked together though the value of the integral
is zero. See Fig. 4.
It was the discovery by Gauss of this very integral, expressing
the work done on a magnetic pole while de
scribing a closed curve in presence of a closed
electric current, and indicating the geometrical
connexion between the two closed curves, that
led him to lament the small progress made in the
Geometry of Position since the time of Leibnitz,
Euler and Vandermonde. We have now, how- Flg> 4>
ever, some progress to report, chiefly due to Riemann, Helmholtz
and Listing.
422.] Let us now investigate the result of integrating with
respect to s round the closed curve.
One of the terms of FT in equation (7) is
f — x dri dz _ di) d A dz^ , .
r3 da- ds ~~ da d£ W ds'
If we now write for brevity
^ f 1 dx 7 „ f 1 dy .. TT f 1 dz
F — I - -r- ds, G = I - -f- ds, R—\- ~ ds, (9)
J r ds J r ds J r ds
the integrals being taken once round the closed curve s, this term
of FT may be written
da- d£ds
and the corresponding term of / n ds will be
da- d£
Collecting all the terms of n, we may now write
This quantity is evidently the rate of decrement of co, the
magnetic potential, in passing along the curve a-, or in other words,
it is the magnetic force in the direction of da:
By assuming da- successively in the direction of the axes of
x, y and z, we obtain for the values of the components of the
magnetic force
42 MAGNETIC SOLENOIDS AND SHELLS. [423-
do> _ dH dG
Ot — ~~~ 7 f. — ~~j ~" T"T~
dt, d-r] d£
d<* _ dF dH
dr] d£ d£
do> _ dG dF
y =~ JT> — ,7 /• ~j
(11)
The quantities F, G, H are the components of the vector-potential
of the magnetic shell whose strength is unity, and whose edge is
the curve s. They are not, like the scalar potential o>, functions
having a series of values, but are perfectly determinate for every
point in space.
The vector-potential at a point P due to a magnetic shell bounded
by a closed curve may be found by the following geometrical
construction :
Let a point Q travel round the closed curve with a velocity
numerically equal to its distance from P, and let a second point
R start from A and travel with a velocity the direction of which
is always parallel to that of Q, but whose magnitude is unity.
When Q has travelled once round the closed curve join AR, then
the line AR represents in direction and in numerical magnitude
the vector-potential due to the closed curve at P.
Potential Energy of a Magnetic Shell placed in a Magnetic Field.
423.] We have already shewn, in Art. 410, that the potential
energy of a shell of strength <£ placed in a magnetic field whose
potential is T9 is
rffidV d7 dY\ 70
x-tJJ ('is +*?+•*)** ^
where I, m, n are the direction-cosines of the normal to the shell
drawn from the positive side, and the surface-integral is extended
over the shell.
Now this surface-integral may be transformed into a line-integral
by means of the vector-potential of the magnetic field, and we
-+cf+^,
where the integration is extended once round the closed curve s
which forms the edge of the magnetic shell, the direction of ds
being opposite to that of the hands of a watch when viewed from
the positive side of the shell.
If we now suppose that the magnetic field is that due to a
423.] POTENTIAL OF TWO CLOSED CURVES. 43
second magnetic shell whose strength is <£', the values of F, G, H
will be
where the integrations are extended once round the curve /, which
forms the edge of this shell.
Substituting these values in the expression for M we find
, ff I fdx dx dy dy dz dz^ .
Jf = —$$'// - (-J- -j-' + ir j' + -j--,,)dsds', (15)
^ JJ r ^ds ds ds ds ds ds'
where the integration is extended once round s and once round /.
This expression gives the potential energy due to the mutual action
of the two shells, and is, as it ought to be, the same when s and /
are interchanged. This expression with its sign reversed, when the
strength of each shell is unity, is called the potential of the two
closed curves s and /. It is a quantity of great importance in the
theory of electric currents. If we write e for the angle between
the directions of the elements ds and ds', the potential of s and /
may be written
(16)
It is evidently a quantity of the dimension of a line.
CHAPTER IV.
INDUCED MAGNETIZATION.
424.] WE have hitherto considered the actual distribution of
magnetization in a magnet as given explicitly among the data
of the investigation. We have not made any assumption as to
whether this magnetization is permanent or temporary, except in
those parts of our reasoning in which we have supposed the magnet
broken up into small portions, or small portions removed from
the magnet in such a way as not to alter the magnetization of
any part.
We have now to consider the magnetization of bodies with
respect to the mode in which it may be produced and changed.
A bar of iron held parallel to the direction of the earth's magnetic
force is found to become magnetic, with its poles turned the op
posite way from those of the earth, or the same way as those of
a compass needle in stable equilibrium.
Any piece of soft iron placed in a magnetic field is found to exhibit
magnetic properties. If it be placed in a part of the field where
the magnetic force is great, as between the poles of a horse-shoe
magnet, the magnetism of the iron becomes intense. If the iron
is removed from the magnetic field, its magnetic properties are
greatly weakened or disappear entirely. If the magnetic properties
of the iron depend entirely on the magnetic force of the field in
which it is placed, and vanish when it is removed from the field,
it is called Soft iron. Iron which is soft in the magnetic sense
is also soft in the literal sense. It is easy to bend it and give
it a permanent set, and difficult to break it.
Iron which retains its magnetic properties when removed from
the magnetic field is called Hard iron. Such iron does not take
425.] SOFT AND HARD STEEL. 45
up the magnetic state so readily as soft iron. The operation of
hammering-, or any other kind of vibration, allows hard iron under
the influence of magnetic force to assume the magnetic state more
readily, and to part with it more readily when the magnetizing
force is removed. Iron which is magnetically hard is also more
stiff to bend and more apt to break.
The processes of hammering, rolling, wire-drawing, and sudden
cooling tend to harden iron, and that of annealing tends to
soften it.
The magnetic as well as the mechanical differences between steel
of hard and soft temper are much greater than those between hard
and soft iron. Soft steel is almost as easily magnetized and de
magnetized as iron, while the hardest steel is the best material
for magnets which we wish to be permanent.
Cast iron, though it contains more carbon than steel, is not
so retentive of magnetization.
If a magnet could be constructed so that the distribution of its
magnetization is not altered by any magnetic force brought to
act upon it, it might be called a rigidly magnetized body. The
only known body which fulfils this condition is a conducting circuit
round which a constant electric current is made to flow.
Such a circuit exhibits magnetic properties, and may therefore be
called an electromagnet, but these magnetic properties are not
affected by the other magnetic forces in the field. We shall return
to this subject in Part IV.
All actual magnets, whether made of hardened steel or of load
stone, are found to be affected by any magnetic force which is
brought to bear upon them.
It is convenient, for scientific purposes, to make a distinction
between the permanent and the temporary magnetization, defining
the permanent magnetization as that which exists independently
of the magnetic force, and the temporary magnetization as that
which depends on this force. We must observe, however, that
this distinction is not founded on a knowledge of the intimate
nature of magnetizable substances : it is only the expression of
an hypothesis introduced for the sake of bringing calculation to
bear on the phenomena. We shall return to the physical theory
of magnetization in Chapter VI.
425.] At present we shall investigate the temporary magnet
ization on the assumption that the magnetization of any particle
of the substance depends solely on the magnetic force acting on
46 INDUCED MAGNETIZATION. [425.
that particle. This magnetic force may arise partly from external
causes, and partly from the temporary magnetization of neigh
bouring particles.
A body thus magnetized in virtue of the action of magnetic
force, is said to be magnetized by induction, and the magnetization
is said to be induced by the magnetizing force.
The magnetization induced by a given magnetizing force differs
in different substances. It is greatest in the purest and softest
iron, in which the ratio of the magnetization to the magnetic force
may reach the value 32, or even 45 *.
Other substances, such as the metals nickel and cobalt, are
capable of an inferior degree of magnetization, and all substances
when subjected to a sufficiently strong magnetic force, are found
to give indications of polarity.
When the magnetization is in the same direction as the magnetic
force, as in iron, nickel, cobalt, &c., the substance is called Para
magnetic, Ferromagnetic, or more simply Magnetic. When the
induced magnetization is in the direction opposite to the magnetic
force, as in bismuth, &c., the substance is said to be Diamagnetic.
In all these substances the ratio of the magnetization to the
magnetic force which produces it is exceedingly small, being only
about — 4 o (H) o Q m the case °f bismuth, which is the most highly
diamagnetic substance known.
In crystallized, strained, and organized substances the direction
of the magnetization does not always coincide with that of the
magnetic force which produces it. The relation between the com
ponents of magnetization, referred to axes fixed in the body, and
those of the magnetic force, may be expressed by a system of three
linear equations. Of the nine coefficients involved in these equa
tions we shall shew that only six are independent. The phenomena
of bodies of this kind are classed under the name of Magnecrystallic
phenomena.
When placed in a field of magnetic force, crystals tend to set
themselves so that the axis of greatest paramagnetic, or of least
diamagnetic, induction is parallel to the lines of magnetic force.
See Art. 435.
In soft iron, the direction of the magnetization coincides with
that of the magnetic force at the point, and for small values of
the magnetic force the magnetization is nearly proportional to it.
* Thaten, Nova Ada, Reg. Soc. Sc., Upsal., 1863.
427.] PROBLEM OF INDUCED MAGNETIZATION. 47
As the magnetic force increases, however, the magnetization in
creases more slowly, and it would appear from experiments described
in Chap. VI, that there is a limiting value of the magnetization,
beyond which it cannot pass, whatever be the value of the
magnetic force.
In the following outline of the theory of induced magnetism,
we shall begin by supposing the magnetization proportional to the
magnetic force, and in the same line with it.
Definition of the Coefficient of Induced Magnetization.
426.] Let $ be the magnetic force, defined as in Art. 398, at
any point of the body, and let 3 be the magnetization at that
point, then the ratio of 3 to § is called the Coefficient of Induced
Magnetization.
Denoting this coefficient by K, the fundamental equation of
induced magnetism is
The coefficient K is positive for iron and paramagnetic substances,
and negative for bismuth and diamagnetic substances. It reaches
the value 32 in iron, and it is said to be large in the case of nickel
and cobalt, but in all other cases it is a very small quantity, not
greater than 0.00001.
The force <£) arises partly from the action of magnets external
to the body magnetized by induction, and partly from the induced
magnetization of the body itself, Both parts satisfy the condition
of having a potential.
427.] Let V be the potential due to magnetism external to the
body, let X2 be that due to the induced magnetization, then if
U is the actual potential due to both causes
u= r+a. (2)
Let the components of the magnetic force «£), resolved in the
directions of x, y, z, be a, /3, y, and let those of the magnetization
3 be A, B, C, then by equation (1),
A = K a,
*=K/3, (3)
C = Ky.
Multiplying these equations by dx, dy, dz respectively, and
adding, we find
Adx + Bdy+Cdz = K(
48 INDUCED MAGNETIZATION. [427.
But since a, (3 and y are derived from the potential U, we may
write the second member —KdU.
Hence, if /c is constant throughout the substance, the first member
must also be a complete differential of a function of #, y and z,
which we shall call $, and the equation becomes
i A d(b „ d(b d(b
where A = -f- , B = ~- , C — — - . (5)
ax dy dz
The magnetization is therefore lamellar, as defined in Art. 412.
It was shewn in Art. 386 that if p is the volume-density of free
magnetism,
(dA dB dC.
P- — (-J- +-J- + T-}'
x## dy dz '
which becomes in virtue of equations (3),
/da d(3 dy\
\lx dy dz'
But, by Art. 77,
da dj3 dy _
dx dy dz ~
Hence (l+47r*)p = 0,
whence p = 0 (6)
throughout the substance, and the magnetization is therefore sole-
noidal as well as lamellar. See Art. 407.
There is therefore no free magnetism except on the bounding
surface of the body. If v be the normal drawn inwards from the
surface, the magnetic surface-density is
d^> (-^
a- = j-- (7)
dv
The potential II due to this magnetization at any point may
therefore be found from the surface-integral
«-//=
dS. (8)
The value of £1 will be finite and continuous everywhere, and
will satisfy Laplace's equation at every point both within and
without the surface. If we distinguish by an accent the value
of H outside the surface, and if v be the normal drawn outwards,
we have at the surface
Of =0.1 (9)
428.] POISSON'S METHOD. 49
da da'
— + ^ = -4™, by Art. 78,
= 4*8.^). , ., -..: : .
dU
= -47rKj;> bF(4)»
fdV d^ ,
= -47rK(^+^),by(2).
We may therefore write the surface-condition
Hence the determination of the magnetism induced in a homo
geneous isotropic body, bounded by a surface S, and acted upon by
external magnetic forces whose potential is V9 may be reduced to
the following mathematical problem.
We must find two functions H and H' satisfying the following
conditions :
Within the surface S9 XI must be finite and continuous, and must
satisfy Laplace's equation.
Outside the surface S, Of must be finite and continuous, it must
vanish at an infinite distance, and must satisfy Laplace's equation.
At every point of the surface itself, H = Of, and the derivatives
of H, Of and V with respect to the normal must satisfy equation
(10). _
This method of treating the problem of induced magnetism is
due to Poisson. The quantity k which he uses in his memoirs is
not the same as *, but is related to it as follows :
47TK(£-l)+3/&= 0. (11)
The coefficient K which we have here used was introduced by
J. Neumann.
428.] The problem of induced magnetism may be treated in a
different manner by introducing the quantity which we have called,
with Faraday, the Magnetic Induction.
The relation between 23, the magnetic induction, «£j, the magnetic
force, and 3> the magnetization, is expressed by the equation
53 = $ + 471 3. (12)
The equation which expresses the induced magnetization in
terms of the magnetic force is
3 = K$. (13)
VOL. IT. E
50 INDUCED MAGNETIZATION. [428.
Hence, eliminating- 3, we find
$ = (1+47TK)£ (14)
as the relation between the magnetic induction and the magnetic
force in substances whose magnetization is induced by magnetic
force.
In the most general case K may be a function, not only of the
position of the point in the substance, but of the direction of the
vector «jp, but in the case which we are now considering K is a
numerical quantity.
If we next write ^ = I + 4 -n K} (15)
we may define /x as the ratio of the magnetic induction to the
magnetic force, and we may call this ratio the magnetic inductive
capacity of the substance, thus distinguishing it from K, the co
efficient of induced magnetization.
If we write U for the total magnetic potential compounded of T7,
the potential due to external causes, and 12 for that due to the
induced magnetization, we may express a, b, c, the components of
magnetic induction, and a, (3, y, the components of magnetic force,
as follows : dU
~}
a = "0 = -M'
dU
e = ™=-*&'j
The components #, d, c satisfy the solenoidal condition
£+!+£=«• (17>
Hence, the potential U must satisfy Laplace's equation
at every point where /ot is constant, that is, at every point within
the homogeneous substance, or in empty space.
At the surface itself, if v is a normal drawn towards the magnetic
substance, and v one drawn outwards, and if the symbols of quan
tities outside the substance are distinguished by accents, the con
dition of continuity of the magnetic induction is
dv , dv dv , dv ,, dv , dv
a-j- +6-j- +0-=- +a'-j- +V -r- +<f -j- = 0; (19)
dx dy dz dx dy dz
429.] FARADAY'S THEORY OF MAGNETIC INDUCTION. 51
or, by equations (16),
fjf, the coefficient of induction outside the magnet, will be unity
unless the surrounding medium be magnetic or diamagnetic.
If we substitute for U its value in terms of V and H, and for
fj> its value in terms of K, we obtain the same equation (10) as we
arrived at by Poisson's method.
The problem of induced magnetism, when considered with respect
to the relation between magnetic induction and magnetic force,
corresponds exactly with the problem of the conduction of electric
currents through heterogeneous media, as given in Art. 309.
The magnetic force is derived from the magnetic potential, pre
cisely as the electric force is derived from the electric potential.
The magnetic induction is a quantity of the nature of a flux,
and satisfies the same conditions of continuity as the electric
current does.
In isotropic media the magnetic induction depends on the mag
netic force in a manner which exactly corresponds with that in
which the electric current depends on the electromotive force.
The specific magnetic inductive capacity in the one problem corre
sponds to the specific conductivity in the other. Hence Thomson,
in his Theory of Induced Magnetism (Reprint, 1872, p. 484), has called
this quantity the permeability of the medium.
We are now prepared to consider the theory of induced magnetism
from what I conceive to be Faraday's point of view.
When magnetic force acts on any medium, whether magnetic or
diamagnetic, or neutral, it produces within it a phenomenon called
Magnetic Induction.
Magnetic induction is a directed quantity of the nature of a flux,
and it satisfies the same conditions of continuity as electric currents
and other fluxes do.
In isotropic media the magnetic force and the magnetic induction
are in the same direction, and the magnetic induction is the product
of the magnetic force into a quantity called the coefficient of
induction, which we have expressed by p.
In empty space the coefficient of induction is unity. In bodies
capable of induced magnetization the coefficient of induction is
1 + 4 TT K = /x, where K is the quantity already defined as the co
efficient of induced magnetization.
429.] Let p, [k be the values of p on opposite sides of a surface
E
52 INDUCED MAGNETIZATION. [4^9-
separating two media, then if F, V are the potentials in the two
media, the magnetic forces towards the surface in the two media
dV , dV'
are -7- and -3-7- •
Av dv
The quantities of magnetic induction through the element of
dV dV
surface dS are u-^-dS and u? -^-j-dS in the two media respect-
r dv dv
ively reckoned towards dS.
Since the total flux towards dS is zero,
dV ,dV
But by the theory of the potential near a surface of density o-,
dV dV
+ 4.47r(r:r= o.
dv dv
Hence -7- (l — —A + 4 TT or = 0.
c?i> V ju, /
If K! is the ratio of the superficial magnetization to the normal
force in the first medium whose coefficient is jot, we have
4 77 KI =
Hence K± will be positive or negative according as /ut is greater
or less than //. If we put ju = 4 TT /c + 1 and p' '= 4 77 / + 1 ,
"47T/+1
In this expression K and K' are the coefficients of induced mag
netization of the first and second medium deduced from experiments
made in air, and KX is the coefficient of induced magnetization of
the first medium when surrounded by the second medium.
If K is greater than K, then /q is negative, or the apparent
magnetization of the first medium is in the opposite direction from
the magnetizing force.
Thus, if a vessel containing a weak aqueous solution of a para
magnetic salt of iron is suspended in a stronger solution of the
same salt, and acted on by a magnet, the vessel moves as if it
were magnetized in the opposite direction from that in which a
magnet would set itself if suspended in the same place.
This may be explained by the hypothesis that the solution in
the vessel is really magnetized in the same direction as the mag
netic force, but that the solution which surrounds the vessel is
magnetized more strongly in the same direction. Hence the vessel
is like a weak magnet placed between two strong ones all mag-
43°-] POISSON'S THEORY OP MAGNETIC INDUCTION. 53
netized in the same direction, so that opposite poles are in contact.
The north pole of the weak magnet points in the same direction
as those of the strong- ones, but since it is in contact with the south
pole of a stronger magnet, there is an excess of south magnetism
in the neighbourhood of its north pole, which causes the small
magnet to appear oppositely magnetized.
In some substances, however, the apparent magnetization is
negative even when they are suspended in what is called a vacuum.
If we assume K = 0 for a vacuum, it will be negative for these
substances. No substance, however, has been discovered for which
K has a negative value numerically greater than — — , and therefore
for all known substances /x is positive.
Substances for which K is negative, and therefore p less than
unity, are called Diamagnetic substances. Those for which K is
positive, and ^ greater than unity, are called Paramagnetic, Ferro
magnetic, or simply magnetic, substances.
We shall consider the physical theory of the diamagnetic and
paramagnetic properties when we come to electromagnetism, Arts.
831-845.
430.] The mathematical theory of magnetic induction was first
given by Poisson *. The physical hypothesis on which he founded
his theory was that of two magnetic fluids, an hypothesis which
has the same mathematical advantages and physical difficulties
as the theory of two electric fluids. In order, however, to explain
the fact that, though a piece of soft iron can be magnetized by
induction, it cannot be charged with unequal quantities of the
two kinds of magnetism, he supposes that the substance in general
is a non-conductor of these fluids, and that only certain small
portions of the substance contain the fluids under circumstances
in which they are free to obey the forces which act on them.
These small magnetic elements of the substance contain each pre
cisely equal quantities of the two fluids, and within each element
the fluids move with perfect freedom, but the fluids can never pass
from one magnetic element to another.
The problem therefore is of the same kind as that relating to
a number of small conductors of electricity disseminated through
a dielectric insulating medium. The conductors may be of any
form provided they are small and do not touch each other.
If they are elongated bodies all turned in the same general
* Memoires de I'lnstitut, 1824.
54 INDUCED MAGNETIZATION. [43O.
direction, or if they are crowded more in one direction than another,
the medium, as Poisson himself shews, will not be isotropic. Poisson
therefore, to avoid useless intricacy, examines the case in which
each magnetic element is spherical, and the elements are dissem
inated without regard to axes. He supposes that the whole volume
of all the magnetic elements in unit of volume of the substance
is k.
We have already considered in Art. 314 the electric conductivity
of a medium in which small spheres of another medium are dis
tributed.
If the conductivity of the medium is ^ , and that of the spheres
ju2, we have found that the conductivity of the composite system is
2)
P = f*l-j
Putting fa = 1 and /ot2 = oc, this becomes
_ 1 + 2/fc
This quantity ju is the electric conductivity of a medium con
sisting of perfectly conducting spheres disseminated through a
medium of conductivity unity, the aggregate volume of the spheres
in unit of volume being k.
The symbol ^ also represents the coefficient of magnetic induction
of a medium, consisting of spheres for which the permeability is
infinite, disseminated through a medium for which it is unity.
The symbol k, which we shall call Poisson's Magnetic Coefficient,
represents the ratio of the volume of the magnetic elements to the
whole volume of the substance.
The symbol K is known as Neumann's Coefficient of Magnet
ization by Induction. It is more convenient than Poisson's.
The symbol ^ we shall call the Coefficient of Magnetic Induction.
Its advantage is that it facilitates the transformation of magnetic
problems into problems relating to electricity and heat.
The relations of these three symbols are as follows :
47TK
3 * =
3*
477
If we put K = 32, the value given by Thalen's* experiments on
* Recherches sur les Proprietes Magnetiques dufer, Nova Ada, Upsal, 1863.
430.] POISSON'S THEORY OF MAGNETIC INDUCTION. 55
soft iron, we find k = |f|-. This, according to Poisson's theory,
is the ratio of the volume of the magnetic molecules to the whole
volume of the iron. It is impossible to pack a space with equal
spheres so that the ratio of their volume to the whole space shall
be so nearly unity, and it is exceedingly improbable that so large
a proportion of the volume of iron is occupied by solid molecules
whatever be their form. This is one reason why we must abandon
Poisson's hypothesis. Others will be stated in Chapter VI. Of
course the value of Poisson's mathematical investigations remains
unimpaired, as they do not rest on his hypothesis, but on the
experimental fact of induced magnetization.
CHAPTER V.
PARTICULAR PROBLEMS IN MAGNETIC INDUCTION.
A Hollow Spherical Shell.
431.] THE first example of the complete solution of a problem
in magnetic induction was that given by Poisson for the case of
a hollow spherical shell acted on by any magnetic forces whatever.
For simplicity we shall suppose the origin of the magnetic forces
to be in the space outside the shell.
If V denotes the potential due to the external magnetic system,
we may expand V in a series of solid harmonics of the form
7= CQ80 + C1S1r + to. + CiSiiA, (1)
where r is the distance from the centre of the shell, #< is a surface
harmonic of order i, and Ci is a coefficient.
This series will be convergent provided r is less than the distance
of the nearest magnet of the system which produces this potential.
Hence, for the hollow spherical shell and the space within it, this
expansion is convergent.
Let the external radius of the shell be a2 and the inner radius alf
and let the potential due to its induced magnetism be H. The form
of the function H will in general be different in the hollow space,
in the substance of the shell, and in the space beyond. If we
expand these functions in harmonic series, then, confining our
attention to those terms which involve the surface harmonic Si9
we shall find that if Q^ is that which corresponds to the hollow
space within the shell, the expansion of Q^ must be in positive har
monics of the form Al St r*, because the potential must not become
infinite within the sphere whose radius is a^.
In the substance of the shell, where r± lies between aL and a2,
the series may contain both positive and negative powers of /*,
of the form
Outside the shell, where r is greater than a2, since the series
HOLLOW SPHERICAL SHELL. 57
must be convergent however great r may be, we must have only
negative powers of /, of the form
The conditions which must be satisfied by the function 12, are :
It must be (1) finite, and (2) continuous, and (3) must vanish at
an infinite distance, and it must (4) everywhere satisfy Laplace's
equation.
On account of (1) Bl = 0.
On account of (2) when r = a^
(4-4,H2i+1-52=0, (2)
and when r = «2,
(^2-J3)^2i+1 + ^2-^3 = 0. (3)
On account of (3) Az = 0, and the condition (4) is satisfied
everywhere, since the functions are harmonic.
But, besides these, there are other conditions to be satisfied at
the inner and outer surface in virtue of equation (10), Art. 427.
At the inner surface where r = alt
, d£l9 d&, dV ,..
<1+4*«>V-ifr+4"'* = <)'
and at the outer surface where r = a2,
d dV
,KN
0.
From these conditions we obtain the equations
iCia12i+l = <), (6)
«22»+1-(^+l)^2)+(^+l)^3+47r^^22i+1=0^ (7)
and if we put
we find
/ /, 2» + l\
4 = -(4™)^ + l)(l-Q) }NtClt (9)
[I a 2t+l^-j
2^+l+477K(^+l)(l-(^) )J^Ci, (10)
(11)
«12i+1)^Ci. (12)
These quantities being substituted in the harmonic expansions
give the part of the potential due to the magnetization of the shell.
The quantity Ni is always positive, since 1 -f 4 ir K can never be
negative. Hence A1 is always negative, or in other words, the
58 MAGNETIC PEOBLEMS. [432.
action of the magnetized shell on a point within it is always op
posed to that of the external magnetic force whether the shell he
paramagnetic or diamagnetic. The actual value of the resultant
potential within the shell is
or (l + 4wjc)(2i+ l^NiCtS.r. (13)
432.] When K is a large number, as it is in the case of soft iron,
then, unless the shell is very thin, the magnetic force within it
is hut a small fraction of the external force.
In this way Sir W. Thomson has rendered his marine galvano
meter independent of external magnetic force hy enclosing it in
a tube of soft iron.
433.] The case of greatest practical importance is that in which
i = 1. In this case
(14)
9(l+47TK)+2(477K)2(l-0')
= -477*13+ 8w«(l — (^) )UViQ, !> (15)
L X dr> —I
£3= 4 7TK(3 + 8 7TK)(#23 — «13)^V1 Ci.
The magnetic force within the hollow shell is in this case uniform
and equal to
9(1+477*)
If we wish to determine K by measuring the magnetic force
within a hollow shell and comparing it with the external magnetic
force, the best value of the thickness of the shell may be found
from the equation
1_
-
2 (4 TT K)2
The magnetic forc"e inside the shell is then half of its value outside.
Since, in the case of iron, K is a number between 20 and 30, the
thickness of the shell ought to be about the hundredth part of its
radius. This method is applicable only when the value of K is
large. When it is very small the value of A^ becomes insensible,
since it depends on the square of K.
434-1 SPHERICAL SHELL. 59
For a nearly solid sphere with a very small spherical hollow,
. 2(4ir«)«
1J
4 77 K
The whole of this investigation might have been deduced directly
from that of conduction through a spherical shell, as given in
Art. 312, by putting ^ = (1 -f 47TK)/£2 in the expressions there given,
remembering that A^ and A2 in the problem of conduction are equi
valent to C1 + A1 and C1 + A2 in the problem of magnetic induction.
434.] The corresponding solution in two dimensions is graphically
represented in Fig. XV, at the end of this volume. The lines of
induction, which at a distance from the centre of the figure are
nearly horizontal, are represented as disturbed by a cylindric rod
magnetized transversely and placed in its position of stable equi
librium. The lines which cut this system at right angles represent
the equipotential surfaces, one of which is a cylinder. The large
dotted circle represents the section of a cylinder of a paramagnetic
substance, and the dotted horizontal straight lines within it, which
are continuous with the external lines of induction, represent the
lines of induction within the substance. The dotted vertical lines
represent the internal equipotential surfaces, and are continuous
with the external system. It will be observed that the lines of
induction are drawn nearer together within the substance, and the
equipotential surfaces are separated farther apart by the paramag
netic cylinder, which, in the language of Faraday, conducts the
lines of induction better than the surrounding medium.
If we consider the system of vertical lines as lines of induction,
and the horizontal system as equipotential surfaces, we have, in
the first place, the case of a cylinder magnetized transversely and
placed in the position of unstable equilibrium among the lines of
force, which it causes to diverge. In the second place, considering
the large dotted circle as the section of a diamagnetic cylinder,
the dotted straight lines within it, together with the lines external
to it, represent the effect of a diamagnetic substance in separating
the lines of induction and drawing together the equipotential
surfaces, such a substance being a worse conductor of magnetic
induction than the surrounding medium.
60 MAGNETIC PROBLEMS. [435-
Case of a Sphere in which the Coefficients of Magnetization are
Different in Different Directions.
435.] Let a, (B, y be the components of magnetic force, and A, £,
C those of the magnetization at any point, then the most general
linear relation between these quantities is given by the equations
A = ^0+^3/3+ q2y, \
£ = q9a+r2p+fly, { (1)
C = p2a+q1h2 + 7-3 y, )
where the coefficients r,jo, q are the nine coefficients of magnet
ization.
Let us now suppose that these are the conditions of magnet
ization within a sphere of radius a, and that the magnetization at
every point of the substance is uniform and in the same direction,
having the components A, 13, C.
Let us also suppose that the external magnetizing force is also
uniform and parallel to one direction, and has for its components
X, Y, Z.
The value of V is therefore
and that of &' the potential of the magnetization outside the sphere is
(3)
The value of H, the potential of the magnetization within the
sphere, is 4-n-
(4)
o
The actual potential within the sphere is V-\- £1, so that we shall
have for the components of the magnetic force within the sphere
a = X — ^TtA, \
0 = 7-J.ir-B, (5)
y =Z-
Hence
+i*r1)^+ twftjjB + iir&
C = &J+ r2Y+frZ, (6)
+(1 +
Solving these equations, we find
A = r/^+K
'' (7)
43^.] CRYSTALLINE SPHERE. 61
where I/ /•/ = r± + ^ TT ( rB rl — p2 q2 4 r-± r2 —
;-A^i)>
&c.,
where D is the determinant of the coefficients on the right side of
equations (6), and D' that of the coefficients on the left.
The new system of coefficients _p' ', /_, / will be symmetrical only
when the system p, q, r is symmetrical, that is, when the co
efficients of the form p are equal to the corresponding ones of
the form q.
436.] The moment of the couple tending to turn the sphere about
the axis of x from y towards z is
f. n Y\\ (Q\
— Jr2 ))* \ /
If we make
X = 0, Y = Fcos 0, Y = Fsin 0,
this corresponds to a magnetic force F in the plane of yz, and
inclined to y at an angle 0. If we now turn the sphere while this
force remains constant the work done in turning the sphere will
T27T
be / LdQ in each complete revolution. But this is equal to
0
Hence, in order that the revolving sphere may not become an
inexhaustible source of energy, j»1/= fa', and similarly j»./= q2 and
These conditions shew that in the original equations the coeffi
cient of B in the third equation is equal to that of C in the second,
and so on. Hence, the system of equations is symmetrical, and the
equations become when referred to the principal axes of mag
netization, TI
A = rr*"i '
C =
(11)
The moment of the couple tending to turn the sphere round the
axis of x is
62 MAGNETIC PROBLEMS. [437-
In most cases the differences between the coefficients of magnet
ization in different directions are very small, so that we may put
This is the force tending to turn a crystalline sphere about the
axis of oo from y towards z. It always tends to place the axis of
greatest magnetic coefficient (or least diamagnetic coefficient) parallel
to the line of magnetic force.
The corresponding case in two dimensions is represented in
Fig. XVI.
If we suppose the upper side of the figure to be towards the
north, the figure represents the lines of force and equipotential
surfaces as disturbed by a transversely magnetized cylinder placed
with the north side eastwards. The resultant force tends to turn
the cylinder from east to north. The large dotted circle represents
a section of a cylinder of a crystalline substance which has a larger
coefficient of induction along an axis from north-east to south-west
than along an axis from north-west to south-east. The dotted lines
within the circle represent the lines of induction and the equipotential
surfaces, which in this case are not at right angles to each other.
The resultant force on the cylinder is evidently to turn it from east
to north.
437.] The case of an ellipsoid placed in a field of uniform and
parallel magnetic force has been solved in a very ingenious manner
by Poisson.
If V is the potential at the point (as, y, z\ due to the gravitation
dV
of a body of any form of uniform density p, then — -=- is the
potential of the magnetism of the same body if uniformly mag
netized in the direction of x with the intensity I = p.
For the value of -- =— 8# at any point is the excess of the value
clx
of V3 the potential of the body, above V, the value of the potential
when the body is moved — §x in the direction of x.
If we supposed the body shifted through the distance — 8#, and
its density changed from p to — p (that is to say, made of repulsive
dV
instead of attractive matter,) then — y-8# would be the potential
due to the two bodies.
Now consider any elementary portion of the body containing a
volume b v. Its quantity is pbv, and corresponding to it there is
437-] ELLIPSOID. 63
an element of the shifted body whose quantity is — pbv at a
distance — 8#. The effect of these two elements is equivalent to
that of a magnet of strength pbr and length 8#. The intensity
of magnetization is found hy dividing the magnetic moment of an
element by its volume. The result is p 8#.
dV
Hence -=- 8# is the magnetic potential of the body magnetized
rl V
with the intensity p bx in the direction of x, and — is that of
ax
the body magnetized with intensity p.
This potential may be also considered in another light. The
body was shifted through the distance — 8# and made of density
—p. Throughout that part of space common to the body in its
two positions the density is zero, for, as far as attraction is con
cerned, the two equal and opposite densities annihilate each other.
There remains therefore a shell of positive matter on one side and
of negative matter on the other, and we may regard the resultant
potential as due to these. The thickness of the shell at a point
where the normal drawn outwards makes an angle e with the axis
of a? is 8 a? cos e and its density is p. The surface-density is therefore
dV
p bx cos 6, and, in the case in which the potential is — , the
surface-density is p cos e.
In this way we can find the magnetic potential of any body
uniformly magnetized parallel to a given direction. Now if this
uniform magnetization is due to magnetic induction, the mag
netizing force at all points within the body must also be uniform
and parallel.
This force consists of two parts, one due to external causes, and
the other due to the magnetization of the body. If therefore the
external magnetic force is uniform and parallel, the magnetic force
due to the magnetization must also be uniform and parallel for
all points within the body.
Hence, in order that this method may lead to a solution of the
clV
problem of magnetic induction, -=- must be a linear function of
doc
the coordinates x, y> z within the body, and therefore V must be
a quadratic function of the coordinates.
Now the only cases with which we are acquainted in which V
is a quadratic function of the coordinates within the body are those
in which the body is bounded by a complete surface of the second
degree, and the only case in which such a body is of finite dimen-
64 MAGNETIC PROBLEMS. [437-
sions is when it is an ellipsoid. We shall therefore apply the
method to the case of an ellipsoid.
be the equation of the ellipsoid, and let 4>0 denote the definite integral
f
'0
Then if we make
dfr
the value of the potential within the ellipsoid will be
70 = - (L x2 + My* + Nz*} + const. (4)
2
If the ellipsoid is magnetized with uniform intensity / in a
direction making angles whose cosines are I, m, n with the axes
of #, y, z, so that the components of magnetization are
A = II, B = Im, C = In,
the potential due to this magnetization within the ellipsoid will be
a = —I(Llx + Mmy + Nnz). (5)
If the external magnetizing force is Ǥ, and if its components
are a, ft, y, its potential will be
r=Xx + Yy + Zz. (6)
The components of the actual magnetizing force at any point
within the body are therefore
X-AL, Y-BM, Z-CN. (7)
The most general relations between the magnetization and the
magnetizing force are given by three linear equations, involving
nine coefficients. It is necessary, however, in order to fulfil the
condition of the conservation of energy, that in the case of magnetic
induction three of these should be equal respectively to other three,
so that we should have
A = K,(X-AL) + Kfs(Y-BM) + K'2(Z-CN},
B = K\ (X-AL) + K2i(Y-BM) + K\(Z-CN], (8)
C = K'2(X-AL) + K\(Y-BM) + Kz(Z-CN}.
From these equations we may determine J, B and C in terms
of X, Y} Z, and this will give the most general solution of the
problem.
The potential outside the ellipsoid will then be that due to the
* See Thomson and Tait's Natural Philosophy, § 522.
438.] ELLIPSOID. 65
magnetization of the ellipsoid together with that due to the external
magnetic force.
438.] The only case of practical importance is that in which
K\ = K2 = K3 = 0. (9)
We have then
If the ellipsoid
flattened form,
A —
"i
X 1
(10)
and is of the planetary or
: (ID
7?
K2
T '
JJ
V
C =
has two
b= c
1+K2M~
K3 g
l+K3N
axes equal,
a
(12)
l-e
M = N = 2 , (±-^ sin-'*- ™) .
\ e* e2 ' J
If the ellipsoid is of the ovary or elongated form
a — b = A/1 — e*c; (13)
In the case of a sphere, when e = 0,
— .«. -^- ^ j
In the case of a very flattened planetoid L becomes in the limit
equal to 4 TT, and M and JV become 7r2 - •
In the case of a very elongated ovoid L and M approximate
to the value 2 TT, while N approximates to the form
a2,, 2c ,
and vanishes when e = 1 .
It appears from these results that —
(1) When K, the coefficient of magnetization, is very small,
whether positive or negative, the induced magnetization is nearly
equal to the magnetizing force multiplied by K, and is almost
independent of the form of the body.
VOL. II. F
66 MAGNETIC PROBLEMS.
(2) When K is a large positive quantity, the magnetization depends
principally on the form of the body,, and is almost independent of
the precise value of /c, except in the case of a longitudinal force
acting on an ovoid so elongated that NK is a small' quantity though
K is large.
(3) If the value of K could be negative and equal to — we
should have an infinite value of the magnetization in the case of
a magnetizing force acting normally to a flat plate or disk. The
absurdity of this result confirms what we said in Art. 428.
Hence, experiments to determine the value of K may be made
on bodies of any form provided K is very small, as it is in the case
of all diamagnetic bodies, and all magnetic bodies except iron,
nickel, and cobalt.
If, however, as in the case of iron, K is a large number, experi
ments made on spheres or flattened figures are not suitable to
determine K ; for instance, in the case of a sphere the ratio of the
magnetization to the magnetizing force is as 1 to 4.22 if K = 30,
as it is in some kinds of iron, and if K were infinite the ratio would
be as 1 to 4.19, so that a very small error in the determination
of the magnetization would introduce a very large one in the
value of K.
But if we make use of a piece of iron in the form of a very
elongated ovoid, then, as long as NK is of moderate value com
pared with unity, we may deduce the value of K from a determination
of the magnetization, and the smaller the value of JV the more
accurate will be the value of K.
In fact, if NK be made small enough, a small error in the value
of N itself will not introduce much error, so that we may use
any elongated body, such as a wire or long rod, instead of an
ovoid.
We must remember, however, that it is only when the product
JV~/c is small compared with unity that this substitution is allowable.
In fact the distribution of magnetism on a long cylinder with flat
ends does not resemble that on a long ovoid, for the free mag
netism is very much concentrated towards the ends of the cylinder,
whereas it varies directly as the distance from the equator in the
case of the ovoid.
The distributi6n of electricity on a cylinder, however, is really
comparable with that on an ovoid, as we have already seen,
Art. 152.
439-] CYLINDER. 67
These results also enable us to understand why the magnetic
moment of a permanent magnet can be made so much greater when
the magnet has an elongated form. If we were to magnetize a
disk with intensity / in a direction normal to its surface, and then
leave it to itself, the interior particles would experience a constant
demagnetizing force equal to 4 TT I, and this, if not sufficient of
itself to destroy part of the magnetization, would soon do so if
aided by vibrations or changes of temperature.
If we were to magnetize a cylinder transversely the demagnet
izing force would be only 2 TT I.
If the magnet were a sphere the demagnetizing force would
be £*/.
In a disk magnetized transversely the demagnetizing force is
a
7T2 - 1) and in an elongated ovoid magnetized longitudinally it
0
a2 2c
is least of all, being 4 TT -^ 7 log ---
G a
Hence an elongated magnet is less likely to lose its magnetism
than a short thick one.
The moment of the force acting on an ellipsoid having different
magnetic coefficients for the three axes which tends to turn it about
the axis of #, is
Hence, if *2 and K3 are small, this force will depend principally
on the crystalline quality of the body and not on its shape, pro
vided its dimensions are not very unequal, but if K2 and *3 are
considerable, as in the case of iron, the force will depend principally
on the shape of the body, and it will turn so as to set its longer
axis parallel to the lines of force.
If a sufficiently strong, yet uniform, field of magnetic force could
be obtained, an elongated isotropic diamagnetic body would also
set itself with its longest dimension parallel to the lines of magnetic
force.
439.] The question of the distribution of the magnetization of
an ellipsoid of revolution under the action of any magnetic forces
has been investigated by J. Neumann*. Kirchhofff has extended
the method to the case of a cylinder of infinite length acted on by
any force.
* Crelle, bd. xxxvii (1848).
t Crelle, bd. xlviii (1854).
F 2
68 MAGNETIC PROBLEMS. [439-
Green, in the 17th section of his Essay, has given an invest
igation of the distribution of magnetism in a cylinder of finite
length acted on by a uniform external force parallel to its axis.
Though some of the steps of this investigation are not very
rigorous, it is probable that the result represents roughly the
actual magnetization in this most important case. It certainly
expresses very fairly the transition from the case of a cylinder
for which K is a large number to that in which it is very small,
but it fails entirely in the case in which K is negative, as in
diamagnetic substances.
Green finds that the linear density of free magnetism at a
distance x from the middle of a cylinder whose radius is a and
whose length is 2 I, is
px
ea +e
where p is a numerical quantity to be found from the equation
0.231863 — 2 \ogep + 2p = - -—
The following are a few of the corresponding values of p and K.
K K
oo 0
336.4 0.01
62.02 0.02
48.416 0.03
29.475 0.04
20.185 0.05
14.794 0.06
11.802 0.07
9.137 0.08
7.517 0.09
6.319 0.10
0.1427 1.00
0.0002 10.00
0.0000 oo
negative imaginary.
When the length of the cylinder is great compared with its
radius, the whole quantity of free magnetism on either side of
the middle of the cylinder is, as it ought to be,
M= v2aKX.
Of this \p M is on the flat end of the cylinder, and the distance
of the centre of gravity of the whole quantity M from the end
a
of the cylinder is -
P
When K is very small p is large, and nearly the whole free
magnetism is on the ends of the cylinder. As K increases p
diminishes, and the free magnetism is spread over a greater distance
44O-] FORCE ON PARA- AND DIA-MAGNETIC BODIES. 69
from the ends. When K is infinite the free magnetism at any
point of the cylinder is simply proportional to its distance from
the middle point, the distribution being similar to that of free
electricity on a conductor in a field of uniform force.
440.] In all substances except iron, nickel, and cobalt, the co
efficient of magnetization is so small that the induced magnetization
of the body produces only a very slight alteration of the forces in
the magnetic field. We may therefore assume, as a first approx
imation, that the actual magnetic force within the body is the same
as if the body had not been there. The superficial magnetization
dV dV
of the body is therefore, as a first approximation, K -j- , where -=-
is the rate of increase of the magnetic potential due to the external
magnet along a normal to the surface drawn inwards. If we
now calculate the potential due to this superficial distribution, we
may use it in proceeding to a second approximation.
To find the mechanical energy due to the distribution of mag
netism on this first approximation we must find the surface-integral
taken over the whole surface of the body. Now we have shewn in
Art. 100 that this is equal to the volume-integral
/*/*/* ~^r~T7 ^ j 77" 2
taken through the whole space occupied by the body, or, if R is the
resultant magnetic force,
E = -
Now since the work done by the magnetic force on the body
during a displacement 8# is Xbos where X is the mechanical force
in the direction of SB, and since
/
= constant,
which shews that the force acting on the body is as if every part
of it tended to move from places where R2 is less to places where
it is greater with a force which on every unit of volume is
rf.JP
K dx '
70 MAGNETIC PEOBLEMS.
If K is negative, as in diamagnetic bodies, this force is, as Faraday
first shewed, from stronger to weaker parts of the magnetic field.
Most of the actions observed in the case of diamagnetic bodies
depend on this property.
Skip's Magnetism.
441.] Almost every part of magnetic science finds its use in
navigation. The directive action of the earth's magnetism on the
compass needle is the only method of ascertaining the ship's course
when the sun and stars are hid. The declination of the needle from
the true meridian seemed at first to be a hindrance to the appli
cation of the compass to navigation, but after this difficulty had
been overcome by the construction of magnetic charts it appeared
likely that the declination itsylf would assist the mariner in de
termining his ship's place.
The greatest difficulty in navigation had always been to ascertain
the longitude ; but since the declination is different at different
points on the same parallel of latitude, an observation of the de
clination together with a knowledge of the latitude would enable
the mariner to find his position on the magnetic chart.
But in recent times iron is so largely used in the construction of
ships that it has become impossible to use the compass at all without
taking into account the action of the ship, as a magnetic body,
on the needle.
To determine the distribution of magnetism in a mass of iron
of any form under the influence of the earth's magnetic force,
even though not subjected to mechanical strain or other disturb
ances, is, as we have seen, a very difficult problem.
In this case, however, the problem is simplified by the following
considerations.
The compass is supposed to be placed with its centre at a fixed
point of the ship, and so far from any iron that the magnetism
of the needle does not induce any perceptible magnetism in the
ship. The size of the compass needle is supposed so small that
we may regard the magnetic force at any point of the needle as
the same.
The iron of the ship is supposed to be of two kinds only.
(1) Hard iron, magnetized in a constant manner.
(2) Soft iron, the magnetization of which is induced by the earth
or other magnets.
In strictness we must admit that the hardest iron is not only
SHIP'S MAGNETISM. 71
capable of induction but that it may lose part of its so-called
permanent magnetization in various ways.
The softest iron is capable of retaining what is called residual
magnetization. The actual properties of iron cannot be accurately
represented by supposing it compounded of the hard iron and the
soft iron above defined. But it has been found that when a ship
is acted on only by the earth's magnetic force, and not subjected
to any extraordinary stress of weather, the supposition that the
magnetism of the ship is due partly to permanent magnetization
and partly to induction leads to sufficiently accurate results when
applied to the correction of the compass.
The equations on which the theory of the variation of the compass
is founded were given by Poisson in the fifth volume of the
Memoires de I'Institut, p. 533 (1824).
The only assumption relative to induced magnetism which is
involved in these equations is, that if a magnetic force X due to
external magnetism produces in the iron of the ship an induced
magnetization, and if this induced magnetization exerts on the
compass needle a disturbing force whose components are JT', Y'9 Z',
then, if the external magnetic force is altered in a given ratio,
the components of the disturbing force will be altered in the
same ratio.
It is true that when the magnetic force acting on iron is very
great the induced magnetization is no longer proportional to the
external magnetic force, but this want of proportionality is quite
insensible for magnetic forces of the magnitude of those due to the
earth's action.
Hence, in practice we may assume that if a magnetic force
whose value is unity produces through the intervention of the iron
of the ship a disturbing force at the compass needle whose com
ponents are a in the direction of #, d in that of y, and g in that of z,
the components of the disturbing force due to a force X in the
direction of x will be aX, dX, and gX.
If therefore we assume axes fixed in the ship, so that x is towards
the ship's head, y to the starboard side, and z towards the keel,
and if X, Y, Z represent the components of the earth's magnetic
force in these directions, and X', Y', Z' the components of the
combined magnetic force of the earth and ship on the compass
needle, X' = X+aX+bY+c Z+P, )
Y' = Y+dX+eY+fZ+Q, (1)
72 MAGNETIC PROBLEMS. [44 1-
In these equations #, #, c, d, e,f, g, h, Jc are nine constant co
efficients depending on the amount, the arrangement, and the
capacity for induction of the soft iron of the ship.
P, Q, and E are constant quantities depending on the permanent
magnetization of the ship.
It is evident that these equations are sufficiently general if
magnetic induction is a linear function of magnetic force, for they
are neither more nor less than the most general expression of a
vector as a linear function of another vector.
It may also be shewn that they are not too general, for, by a
proper arrangement of iron, any one of the coefficients may be
made to vary independently of the others.
Thus, a long thin rod of iron under the action of a longitudinal
magnetic force acquires poles, the strength of each of which is
numerically equal to the cross section of the rod multiplied by
the magnetizing force and by the coefficient of induced magnet
ization. A magnetic force transverse to the rod produces a much
feebler magnetization, the effect of which is almost insensible at
a distance of a few diameters.
If a long iron rod be placed fore and aft with one end at a
distance x from the compass needle, measured towards the ship's
head, then, if the section of the rod is A, and its coefficient of
magnetization K, the strength of the pole will be A K X, and, if
A = — — , the force exerted by this pole on the compass needle
will be aX. The rod may be supposed so long that the effect of
the other pole on the compass may be neglected.
We have thus obtained the means of giving any required value
to the coefficient a.
If we place another rod of section B with one extremity at the
same point, distant x from the compass toward the head of the
vessel, and extending to starboard to such a distance that the
distant pole produces no sensible effect on the compass, the dis
turbing force due to this rod will be in the direction of x, and
B K.Y bx*
equal to — x - , or if B = , the force will be b Y.
X2 K '
This rod therefore introduces the coefficient b.
A third rod extending downwards from the same point will
introduce the coefficient <?.
The coefficients d, e,f may be produced by three rods extending
to head, to starboard, and downward from a point to starboard of
44i.] SHIP'S MAGNETISM. 73
the compass, and g, h, k by three rods in parallel directions from
a point below the compass.
Hence each of the nine coefficients can be separately varied by
means of iron rods properly placed.
The quantities P, Q, R are simply the components of the force
on the compass arising from the permanent magnetization of the
ship together with that part of the induced magnetization which
is due to the action of this permanent magnetization.
A complete discussion of the equations (1), and of the relation
between the true magnetic course of the ship and the course as
indicated by the compass, is given by Mr. Archibald Smith in the
Admiralty Manual of the Deviation of the Compass.
A valuable graphic method of investigating the problem is there
given. Taking a fixed point as origin, a line is drawn from this
point representing in direction and magnitude the horizontal part
of the actual magnetic force on the compass-needle. As the ship
is swung round so as to bring her head into different azimuths
in succession, the extremity of this line describes a curve, each
point of which corresponds to a particular azimuth.
Such a curve, by means of which the direction and magnitude of
the force on the compass is given in terms of the magnetic course
of the ship, is called a Dygogram.
There are two varieties of the Dygogram. In the first, the curve
is traced on a plane fixed in space as the ship turns round. In
the second kind, the curve is traced on a plane fixed with respect
to the ship.
The dygogram of the first kind is the Lima9on of Pascal, that
of the second kind is an ellipse. For the construction and use of
these curves, and for many theorems as interesting to the mathe
matician as they are important to the navigator, the reader is
referred to the Admiralty Manual of the Deviation of the Compass.
CHAPTER VI.
WEBER'S THEORY OF INDUCED MAGNETISM.
442.] WE have seen that Poisson supposes the magnetization of
iron to consist in a separation of the magnetic fluids within each
magnetic molecule. If we wish to avoid the assumption of the
existence of magnetic fluids, we may state the same theory in
another form, hy saying that each molecule of the iron, when the
magnetizing force acts on it, becomes a magnet.
Weber's theory differs from this in assuming that the molecules
of the iron are always magnets, even before the application of
the magnetizing force, but that in ordinary iron the magnetic
axes of the molecules are turned indifferently in every direction,
so that the iron as a whole exhibits no magnetic properties.
When a magnetic force acts on the iron it tends to turn the
axes of the molecules all in one direction, and so to cause the iron,
as a whole, to become a magnet.
If the axes of all the molecules were set parallel to each other,
the iron would exhibit the greatest intensity of magnetization of
which it is capable. Hence Weber's theory implies the existence
of a limiting intensity of magnetization, and the experimental
evidence that such a limit exists is therefore necessary to the
theory. Experiments shewing an approach to a limiting value of
magnetization have been made by Joule * and by J. Miiller f.
The experiments of Beetz J on electrotype iron deposited under
the action of magnetic force furnish the most complete evidence
of this limit, —
A silver wire was varnished, and a very narrow line on the
* Annals of Electricity, iv. p. 131, 1839 ; Phil Mag. [4] ii. p. 316.
t Pogg., Ann. Ixxix. p. 337, 1850.
+ Pogg. cxi. 1860.
443-] THE MOLECULES OF IRON ARE MAGNETS. 75
metal was laid bare by making1 a fine longitudinal scratch on the
varnish. The wire was then immersed in a solution of a salt of
iron, and placed in a magnetic field with the scratch in the direction
of a line of magnetic force. By making the wire the cathode of
an electric current through the solution, iron was deposited on
the narrow exposed surface of the wire, molecule by molecule. The
filament of iron thus formed was then examined magnetically. Its
magnetic moment was found to be very great for so small a mass
of iron, and when a powerful magnetizing force was made to act
in the same direction the increase of temporary magnetization was
found to be very small, and the permanent magnetization was not
altered. A magnetizing force in the reverse direction at once
reduced the filament to the condition of iron magnetized in the
ordinary way.
Weber's theory, which supposes that in this case the magnetizing
force placed the axis of each molecule in the same direction during
the instant of its deposition, agrees very well with what is
observed.
Beetz found that when the electrolysis is continued under the
action of the magnetizing force the intensity of magnetization
of the subsequently deposited iron diminishes. The axes of the
molecules are probably deflected from the line of magnetizing
force when they are being laid down side by side with the mole
cules already deposited, so that an approximation to parallelism.
can be obtained only in the case of a very thin filament of iron.
If, as Weber supposes, the molecules of iron are already magnets,
any magnetic force sufficient to render their axes parallel as they
are electrolytically deposited will be sufficient to produce the highest
intensity of magnetization in the deposited filament.
If, on the other hand, the molecules of iron are not magnets,
but are only capable of magnetization, the magnetization of the
deposited filament will depend on the magnetizing force in the
same way in which that of soft iron in general depends on
it. The experiments of Beetz leave no room for the latter hy
pothesis.
443.] We shall now assume, with Weber, that in every unit of
volume of the iron there are n magnetic molecules, and that the
magnetic moment of each is m. If the axes of all the molecules
were placed parallel to one another, the magnetic moment of the
unit of volume would be
M = n m,
76 WEBER'S THEORY OF INDUCED MAGNETISM. [443.
and this would be the greatest intensity of magnetization of which
the iron is capable.
In the unmagnetized state of ordinary iron Weber supposes the
axes of its molecules to be placed indifferently in all directions.
To express this, we may suppose a sphere to be described, and
a radius drawn from the centre parallel to the direction of the axis
of each of the n molecules. The distribution of the extremities of
these radii will express that of the axes of the molecules. In
the case of ordinary iron these n points are equally distributed
over every part of the surface of the sphere, so that the number
of molecules whose axes make an angle less than a with the axis
of x is n .
- (I - cos a),
and the number of molecules whose axes make angles with that
of ^, between a and a-f da is therefore
n . j
- sin a a a.
2t
This is the arrangement of the molecules in a piece of iron which
has never been magnetized.
Let us now suppose that a magnetic force X is made to act
on the iron in the direction of the axis of a?, and let us consider
a molecule whose axis was originally inclined a to the axis of so.
If this molecule is perfectly free to turn, it will place itself with
its axis parallel to the axis of a?, and if all the molecules did so,
the very slightest magnetizing force would be found sufficient
to develope the very highest degree of magnetization. This, how
ever, is not the case.
The molecules do not turn with their axes parallel to a?, and
this is either because each molecule is acted on by a force tending
to preserve it in its original direction, or because an equivalent
effect is produced by the mutual action of the entire system of
molecules.
Weber adopts the former of these suppositions as the simplest,
and supposes that each molecule, when deflected, tends to return
to its original position with a force which is the same as that
which a magnetic force D, acting in the original direction of its
axis, would produce.
The position which the axis actually assumes is therefore in the
direction of the resultant of X and D.
Let APB represent a section of a sphere whose radius represents,
on a certain scale, the force D.
443-] DEFLEXION OF AXES OF MOLECULES. 77
Let the radius OP be parallel to the axis of a particular molecule
in its original position.
Let SO represent on the same scale the magnetizing force X
which is supposed to act from 8 towards 0. Then, if the molecule
is acted on by the force X in the direction SO, and by a force
D in a direction parallel to OP, the original direction of its axis,
its axis will set itself in the direction SP, that of the resultant
of X and D.
Since the axes of the molecules are originally in all directions,
P may be at any point of the sphere indifferently. In Fig. 5, in
which X is less than D, SP, the final position of the axis, may be
in any direction whatever, but not indifferently, for more of the
molecules will have their axes turned towards A than towards JS.
In Fig. 6, in which X is greater than D, the axes of the molecules
will be all confined within the cone STT' touching the sphere.
Fig. 5.
Hence there are two different cases according as X is less or
greater than D.
Let a = AOP, the original inclination of the axis of a molecule
to the axis of x.
0 = ASP, the inclination of the axis when deflected by
the force X.
(3 = SPO, the angle of deflexion.
SO = X, the magnetizing force.
OP = D, the force tending towards the original position.
SP = R, the resultant of X and D.
m = magnetic moment of the molecule.
Then the moment of the statical couple due to X, tending to
diminish the angle 0, is
mL = mX sin#,
and the moment of the couple due to D, tending to increase 6, is
mL —
78 WEBER'S THEORY OF INDUCED MAGNETISM. [443.
Equating these values, and remembering that /3 = a — 0, we find
J)sina
tan0 = - -- (1)
X +D cos a
to determine the direction of the axis after deflexion.
We have next to find the intensity of magnetization produced
in the mass by the force X, and for this purpose we must resolve
the magnetic moment of every molecule in the direction of #, and
add all these resolved parts.
The resolved part of the moment of a molecule in the direction
of x is m cos 0.
The number of molecules whose original inclinations lay between
a and a -{-da is % .
-smaaa.
2
We have therefore to integrate
/= f* — cos 6 tin a da, (2)
JQ 2
'remembering that 0 is a function of a.
We may express both 9 and a in terms of JR, and the expression
to be integrated becomes
' (3)
the general integral of which is
In the first case, that in which X is less than D, the limits of
integration are R = D + X and R = D— X. In the second case,
in which X is greater than D, the limits are R = X+ D and
R = X-D.
When X is less than D, I = | ~X. (5)
2
When X is equal to D, I = -mn. (6)
3
1 712
When X is greater than D, I — mn(\ -- — ) ; (7)
* o J\. I
and when X becomes infinite / = mn. (8)
According to this form of the theory, which is that adopted
by Weber *, as the magnetizing force increases from 0 to D, the
* There is some mistake in the formula given by Weber (Trans. Acad. Sax. i.
p. 572 (1852), or Pogg., Ann. Ixxxvii. p. 167 (1852)) as the result of this integration,
the steps of which are not given by him. His formula is
444-] L1MIT OF MAGNETIZATION. 79
magnetization increases in the same proportion. When the mag
netizing force attains the value D, the magnetization is two-thirds
of its limiting value. When the magnetizing force is further
increased, the magnetization, instead of increasing indefinitely,
tends towards a finite limit.
D 2D 3D 4D
Fig. 7.
The law of magnetization is expressed in Fig. 7, where the mag
netizing force is reckoned from 0 towards the right and the mag
netization is expressed by the vertical ordinates. Weber's own
experiments give results in satisfactory accordance with this law.
It is probable, however, that the value of D is not the same for
all the molecules of the same piece of iron, so that the transition
from the straight line from 0 to E to the curve beyond E may not
be so abrupt as is here represented.
444.] The theory in this form gives no account of the residual
magnetization which is found to exist after the magnetizing force
is removed. I have therefore thought it desirable to examine the
results of making a further assumption relating to the conditions
under which the position of equilibrium of a molecule may be
permanently altered.
Let us suppose that the axis" of a magnetic molecule, if deflected
through any angle /3 less than /30, will return to its original
position when the deflecting force is removed, but that if the
deflexion j3 exceeds ^0, then, when the deflecting force is removed,
the axis will not return to its original position, but will be per
manently deflected through an angle /3 — j30, which may be called
the permanent set of the molecule.
This assumption with respect to the law of molecular deflexion
is not to be regarded as founded on any exact knowledge of the
intimate structure of bodies, but is adopted, in our ignorance of
the true state of the case, as an assistance to the imagination in
following out the speculation suggested by Weber.
Let L = Dsin /30, (9)
80 WEBER'S THEORY OF INDUCED MAGNETISM. [444.
then, if the moment of the couple acting on a molecule is less than
ml/, there will be no permanent deflexion, but if it exceeds mL
there will be a permanent change of the position of equilibrium.
To trace the results of this supposition, describe a sphere whose
centre is 0 and radius OL = L.
As long as X is less than L everything will be the same as
in the case already considered, but as soon as X exceeds L it will
begin to produce a permanent deflexion of some of the molecules.
Let us take the case of Fig. 8, in which X is greater than L
but less than D. Through S as vertex draw a double cone touching
the sphere L. Let this cone meet the sphere D in P and Q. Then
if the axis of a molecule in its original position lies between OA
and OP, or between OB and OQ, it will be deflected through an
angle less than /30, and will not be permanently deflected. But if
Fig. 8. Fig. 9.
the axis of the molecule lies originally between OP and OQ, then
a couple whose moment is greater than L will act upon it and
will deflect it into the position SP, and when the force X ceases
to act it will not resume its original direction, but will be per
manently set in the direction OP.
Let us put
L = Xsin00 when 0 = PSA or QSB,
then all those molecules whose axes, on the former hypotheses,
would have values of 6 between 00 and TT — 00 will be made to have
the value 00 during the action of the force X.
During the action of the force X, therefore, those molecules
whose axes when deflected lie within either sheet of the double
cone whose semivertical angle is 00 will be arranged as in the
former case, but all those whose axes on the former theory would
lie outside of these sheets will be permanently deflected, so that
their axes will form a dense fringe round that sheet of the cone
which lies towards A.
445-] MODIFIED THEORY. 81
As X increases, the number of molecules belonging to the cone
about B continually diminishes, and when X becomes equal to D
all the molecules have been wrenched out of their former positions
of equilibrium, and have been forced into the fringe of the cone
round A, so that when X becomes greater than D all the molecules
form part of the cone round A or of its fringe.
When the force X is removed, then in the case in which X is
less than L everything returns to its primitive state. When X
is between L and D then there is a cone round A whose angle
AOP = 00 + /30,
and another cone round B whose angle
BOQ = 00-/30.
Within these cones the axes of the molecules are distributed
uniformly. But all the molecules, the original direction of whose
axes lay outside of both these cones, have been wrenched from their
primitive positions and form a fringe round the cone about A.
If X is greater than D, then the cone round B is completely
dispersed, and all the molecules which formed it are converted into
the fringe round A, and are inclined at the angle 00-f-/30.
445.] Treating this case in the same way as before, we find
for the intensity of the temporary magnetization during the action
of the force X, which is supposed to act on iron which has never
before been magnetized,
When X is less than L, I = - M -_- •
3 J-f
When X is equal to It, I = - M -=j •
When X is between L and 2),
When X is equal to D,
'
When X is greater than D>
When X is infinite, I = M.
When X is less than L the magnetization follows the former
law, and is proportional to the magnetizing force. As soon as X
exceeds L the magnetization assumes a more rapid rate of increase
VOL. n. G
82 WEBER'S THEORY OF INDUCED MAGNETISM. [445.
on account of the molecules beginning to be transferred from the
one cone to the other. This rapid increase, however, soon conies
to an end as the number of molecules forming the negative cone
diminishes, and at last the magnetization reaches the limiting
value M.
If we were to assume that the values of L and of D are different
for different molecules, we should obtain a result in which the
different stages of magnetization are not so distinctly marked.
The residual magnetization, /', produced by the magnetizing force
X, and observed after the force has been removed, is as follows :
When X is less than I/, No residual magnetization.
When X is between L and D,
When X is equal to D,
T2 2
When X is greater than D,
'-J
When X is infinite,
If we make
M = 1000, L = 3, .# = 5,
we find the following values of the temporary and the residual
magnetization : —
Magnetizing Temporary Residual
Force. Magnetization. Magnetization.
x i r
000
1 133 0
2 267 0
3 400 0
4 729 280
5 837 410
6 864 485
7 882 537
8 897 574
oo 1000 810
446.] TEMPORARY AND RESIDUAL MAGNETIZATION. 83
These results are laid down in Fig. 10.
10
I 2 3 4 5 6 7 8 J
JHcufn.etizin.tp jforce
Fig. 10.
The curve of temporary magnetization is at first a straight line
from X = 0 to X = L. It then rises more rapidly till X = 1),
and as X increases it approaches its horizontal asymptote.
The curve of residual magnetization begins when X = _Z/, and
approaches an asymptote at a distance = .8lJf.
It must be remembered that the residual magnetism thus found
corresponds to the case in which, when the external force is removed,
there is no demagnetizing force arising from the distribution of
magnetism in the body itself. The calculations are therefore
applicable only to very elongated bodies magnetized longitudinally.
In the case of short, thick bodies the residual magnetism will be
diminished by the reaction of the free magnetism in the same
way as if an external reversed magnetizing force were made to
act upon it.
446.] The scientific value of a theory of this kind, in which we
make so many assumptions, and introduce so many adjustable
constants, cannot be estimated merely by its numerical agreement
with certain sets of experiments. If it has any value it is because
it enables us to form a mental image of what takes place in a
piece of iron during magnetization. To test the theory, we shall
apply it to the case in which a piece of iron, after being subjected
to a magnetizing force XQ> is again subjected to a magnetizing
force X1.
If the new force X± acts in the same direction in which X0 acted,
which we shall call the positive direction, then, if X± is less than
X^9 it will produce no permanent set of the molecules, and when
X1 is removed the residual magnetization will be the same as
G 2
84: WEBER'S THEORY OF INDUCED MAGNETISM. [446.
that produced by X0 . If Xl is greater than X0 , then it will produce
exactly the same effect as if X0 had not acted.
But let us suppose Xl to act in the negative direction, and let us
suppose XQ = L cosec 00, and Xl = — I/cosec01.
As X1 increases numerically, 0: diminishes. The first molecules
on which X1 will produce a permanent deflexion are those which
form the fringe of the cone round A, and these have an inclination
when undeflected of 00 + J30 .
As soon as 61 — /30 becomes less than 00-f~/30 the process of de
magnetization will commence. Since, at this instant, ^ = 00-f 2^30,
X13 the force required to begin the demagnetization, is less than
XQ, the force which produced the magnetization.
If the value of D and of L were the same for all the molecules,
the slightest increase of X1 would wrench the whole of the fringe
of molecules whose axes have the inclination 00 + /30 into a position
in which their axes are inclined 01 + )30 to the negative axis OB.
Though the demagnetization does not take place in a manner
so sudden as this, it takes place so rapidly as to afford some
confirmation of this mode of explaining the process.
Let us now suppose that by giving a proper value to the reverse
force Xj we have exactly demagnetized the piece of iron.
The axes of the molecules will not now be arranged indiffer
ently in all directions, as in a piece of iron which has never been
magnetized, but will form three groups.
(1) Within a cone of semiangle 01— /30 surrounding the positive
pole, the axes of the molecules remain in their primitive positions.
(2) The same is the case within a cone of semiangle 00— /30
surrounding the negative pole.
(3) The directions of the axes of all the other molecules form
a conical sheet surrounding the negative pole, and are at an
inclination 0l + /30 .
When X0 is greater than D the second group is absent. When
•Xj_ is greater than I) the first group is also absent.
The state of the iron, therefore, though apparently demagnetized,
is in a different state from that of a piece of iron which has never
been magnetized.
To shew this, let us consider the effect of a magnetizing force
X2 acting in either the positive or the negative direction. The
first permanent effect of such a force will be on the third group
of molecules, whose axes make angles = 01 + /30 with the negative
axis.
447-1 MAGNETISM AND TORSION, 85
If the force X2 acts in the negative direction it will begin to
produce a permanent effect as soon as 02 + /30 becomes less than
^i + A)5 that is, as soon as X2 becomes greater than XL. But if
X2 acts in the positive direction it will begin to remagnetize the
iron as soon as 02 — {3 becomes less than Oi + P0, that is, when
02 = Ol -j- 2/30, or while X2 is still much less than X±.
It appears therefore from our hypothesis that —
When a piece of iron is magnetized by means of a force X0i its
magnetism cannot be increased without the application of a force
greater than X0. A reverse force, less than Jf0, is sufficient to
diminish its magnetization.
If the iron is exactly demagnetized by a reversed force X19 then
it cannot be magnetized in the reversed direction without the
application of a force greater than X1} but a positive force less than
Xx is sufficient to begin to remagnetize the iron in its original
direction.
These results are consistent with what has been actually observed
by Ritchie*. Jacobi f, Marianini J, and Joule §.
A very complete account of the relations of the magnetization
of iron and steel to magnetic forces and to mechanical strains is
given by Wiedemann in his Galvanismus. By a detailed com
parison of the effects of magnetization with those of torsion, he
shews that the ideas of elasticity and plasticity which we derive
from experiments on the temporary and permanent torsion of wires
can be applied with equal propriety to the temporary and permanent
magnetization of iron and steel.
447.] Matteucci || found that the extension of a hard iron bar
during the action of the magnetizing force increases its temporary
magnetism. This has been confirmed by Wertheim. In the case
of soft bars the magnetism is diminished by extension.
The permanent magnetism of a bar increases when it is extended,
and diminishes when it is compressed.
Hence, if a piece of iron is first magnetized in one direction,
and then extended in another direction, the direction of magnet
ization will tend to approach the direction of extension. If it be
compressed, the direction of magnetization will tend to become
normal to the direction of compression.
This explains the result of an experiment of Wiedemann's. A
* Phil. Mag., 1833. t Pog., Ann., 1834.
J Ann. de Chimie d de Physique, 1846. § Phil. Trans., 1855, p. 287.
|| Ann. de Chimie et de Physique, 1858.
86
WEBER S THEORY OF INDUCED MAGNETISM.
[448.
current was passed downward through a vertical wire. If, either
during the passage of the current or after it has ceased, the wire
be twisted in the direction of a right-handed screw, the lower end
becomes a north pole.
Fi.
Here the downward current magnetizes every part of the wire
in a tangential direction, as indicated by the letters NS.
The twisting of the wire in the direction of a right-handed screw
causes the portion ABCD to be extended along the diagonal AC
and compressed along the diagonal BD. The direction of magnet
ization therefore tends to approach AC and to recede from BD,
and thus the lower end becomes a north pole and the upper end
a south pole.
Effect of Magnetization on the Dimensions of the Magnet.
448.] Joule *, in 1842, found that an iron bar becomes length
ened when it is rendered magnetic by an electric current in a
coil which surrounds it. He afterwards f shewed, by placing the
bar in water within a glass tube, that the volume of the iron is
not augmented by this magnetization, and concluded that its
transverse dimensions were contracted.
Finally, he passed an electric current through the axis of an iron
tube, and back outside the tube, so as to make the tube into a
closed magnetic solenoid, the magnetization being at right angles
to the axis of the tube. The length of the axis of the tube was
found in this case to be shortened.
He found that an iron rod under longitudinal pressure is also
elongated when it is magnetized. When, however, the rod is
under considerable longitudinal tension, the effect of magnetization
is to shorten it.
* Sturgeon's Annals of Electricity, vol. viii. p. 219.
t Phil. Mag., 1847.
448-] CHANGE OF FORM. 87
This was the case with a wire of a quarter of an inch diameter
when the tension exceeded 600 pounds weight.
In the case of a hard steel wire the effect of the magnetizing
force was in every case to shorten the wire, whether the wire was
under tension or pressure. The change of length lasted only as
long as the magnetizing force was in action, no alteration of length
was observed due to the permanent magnetization of the steel.
Joule found the elongation of iron wires to be nearly proportional
to the square of the actual magnetization, so that the first effect
of a demagnetizing current was to shorten the wire.
On the other hand, he found that the shortening effect on wires
under tension, and on steel, varied as the product of the magnet
ization and the magnetizing current.
Wiedemann found that if a vertical wire is magnetized with its
north end uppermost, and if a current is then passed downwards
through the wire, the lower end of the wire, if free, twists in the
direction of the hands of a watch as seen from above, or, in other
words, the wire becomes twisted like a right-handed screw.
In this case the magnetization due to the action of the current
on the previously existing magnetization is in the direction of
a left-handed screw round the wire. Hence the twisting would
indicate that when the iron is magnetized it contracts in the
direction of magnetization and expands in directions at right angles
to the magnetization. This, however, peems not to agree with Joule's
results.
For further developments of the theory of magnetization, see
Arts. 832-845.
CHAPTER VII.
MAGNETIC MEASUREMENTS.
449.] THE principal magnetic measurements are the determination
of the magnetic axis and magnetic moment of a magnet, and that
of the direction and intensity of the magnetic force at a given
place.
Since these measurements are made near the surface of the earth,
the magnets are always acted on by gravity as well as by terrestrial
magnetism, and since the magnets are made of steel their mag
netism is partly permanent and partly induced. The permanent
magnetism is altered by changes of temperature, by strong in
duction, and by violent blows ; the induced magnetism varies with
every variation of the external magnetic force.
The most convenient way of observing the force acting on a
magnet is by making the magnet free to turn about a vertical
axis. In ordinary compasses this is done by balancing the magnet
on a vertical pivot. The finer the point of the pivot the smaller
is the moment of the friction which interferes with the action of
the magnetic force. For more refined observations the magnet
is suspended by a thread composed of a silk fibre without twist,
either single, or doubled on itself a sufficient number of times, and
so formed into a thread of parallel fibres, each of which supports
as nearly as possible an equal part of the weight. The force of
torsion of such a thread is much less than that of a metal wire
of equal strength, and it may be calculated in terms of the ob
served azimuth of the magnet, which is not the case with the force
arising from the friction of a pivot.
The suspension fibre can be raised or lowered by turning a
horizontal screw which works in a fixed nut. The fibre is wound
round the thread of the screw, so that when the screw is turned
the suspension fibre always hangs in the same vertical line.
450.]
SUSPENSION".
89
The suspension fibre carries a small horizontal divided circle
called the Torsion-circle, and a stirrup with an index, which can
be placed so that the index coincides with any given division of
the torsion circle. The stirrup is so shaped that the magnet bar
can be fitted into it with its axis horizontal, and with any one
of its four sides uppermost.
To ascertain the zero of torsion a non-magnetic body of the
same weight as the magnet is placed
in the stirrup, and the position of
the torsion circle when in equilibrium
ascertained.
The magnet itself is a piece of
hard-tempered steel. According to
Gauss and Weber its length ought
to be at least eight times its greatest
transverse dimension. This is neces
sary when permanence of the direc
tion of the magnetic axis within the
magnet is the most important con
sideration. Where promptness of
motion is required the magnet should
be shorter, and it may even be ad
visable in observing sudden altera
tions in magnetic force to use a bar
magnetized transversely and sus
pended with its longest dimension
vertical *.
450.1 The magnet is provided with
an arrangement for ascertaining its
angular position. For ordinary pur
poses its ends are pointed, and a
divided circle is placed below the
Fig. 13.
ends, by which their positions are read oif by an eye placed in a
plane through the suspension thread and the point of the needle.
For more accurate observations a plane mirror is fixed to the
magnet, so that the normal to the mirror coincides as nearly as
possible with the axis of magnetization. This is the method
adopted by Gauss and Weber.
Another method is to attach to one end of the magnet a lens and
to the other end a scale engraved on glass, the distance of the lens
* Joule, Proc. Phil. Soc., Manchester, Nov. 29, 1864.
90 MAGNETIC MEASUREMENTS. [45O.
from the scale being1 equal to tlie principal focal length of the lens.
The straight line joining the zero of the scale with the optical
centre of the lens ought to coincide as nearly as possible with
the magnetic axis.
As these optical methods of ascertaining the angular position
of suspended apparatus are of great importance in many physical
researches, we shall here consider once for all their mathematical
theory.
Theory of the Mirror Method.
We shall suppose that the apparatus whose angular position is
to be determined is capable of revolving about a vertical axis.
This axis is in general a fibre or wire by which it is suspended.
The mirror should be truly plane, so that a scale of millimetres
may be seen distinctly by reflexion at a distance of several metres
from the mirror.
The normal through the middle of the mirror should pass through
the axis of suspension, and should be accurately horizontal. We
shall refer to this normal as the line of collimation of the ap
paratus.
Having roughly ascertained the mean direction of the line of
collimation during the experiments which are to be made, a tele
scope is erected at a convenient distance in front of the mirror, and
a little above the level of the mirror.
The telescope is capable of motion in a vertical plane, it is
directed towards the suspension fibre just above the mirror, and
a fixed mark is erected in the line of vision, at a horizontal distance
from the object glass equal to twice the distance of the mirror
from the object glass. The apparatus should, if possible, be so
arranged that this mark is on a wall or other fixed object. In
order to see the mark and the suspension fibre at the same time
through the telescope, a cap may be placed over the object glass
having a slit along a vertical diameter. This should be removed
for the other observations. The telescope is then adjusted so that
the mark is seen distinctly to coincide with the vertical wire at the
focus of the telescope. A plumb-line is then adjusted so as to
pass close in front of the optical centre of the object glass and
to hang below the telescope. Below the telescope and just behind
the plumb-line a scale of equal parts is placed so as to be bisected
at right angles by the plane through the mark, the suspension-fibre,
and the plumb-lino. The sum of the heights of the scale and the
450.]
THE MIRROR METHOD.
91
object glass should be equal to twice the height of the mirror from
the floor. The telescope being now directed towards the mirror
will see in it the reflexion of the scale. If the part of the scale
where the plumb-line crosses it appears to coincide with the vertical
wire of the telescope, then the line of collimation of the mirror
coincides with the plane through the mark and the optical centre
of the object glass. If the vertical wire coincides with any other
division of the scale, the angular position of the line of collimation
is to be found as follows : —
Let the plane of the paper be horizontal, and let the various
points be projected on this plane. Let 0 be the centre of the
object glass of the telescope, P the fixed mark, P and the vertical
wire of the telescope are conjugate foci with respect to the object
glass. Let M be the point where OP cuts the plane of the mirror.
Let MN be the normal to the mirror ; then OMN = 6 is the angle
which the line of collimation makes with the fixed plane. Let MS
be a line in the plane of OM and MN, such that NMS = OMN,
then S will be the part of the scale which will be seen by reflexion
to coincide with the vertical wire of the telescope. Now, since
X
X
xx ---'V
Fig. 14.
MN is horizontal, the projected angles OMN and NMS in the
figure are equal, and QMS =20. Hence OS = OMtan.20.
We have therefore to measure OM in terms of the divisions of
the scale ; then, if s0 is the division of the scale which coincides with
the plumb-line, and s the observed division,
whence 6 may be found. In measuring OM we must remember
that if the mirror is of glass, silvered at the back, the virtual image
of the reflecting surface is at a distance behind the front surface
92
MAGNETIC MEASUREMENTS.
[450.
of the glass = — , where t is the thickness of the glass, and //, is
the index of refraction.
We must also remember that if the line of suspension does not
pass through the point of reflexion, the position of M will alter
with 0. Hence, when it is possible, it is advisable to make the
centre of the mirror coincide with the line of suspension.
It is also advisable, especially when large angular motions have
to be observed, to make the scale in the form of a concave cylindric
surface, whose axis is the line of suspension. The angles are then
observed at once in circular measure without reference to a table
of tangents. The scale should be carefully adjusted, so that the
axis of the cylinder coincides with the suspension fibre. The
numbers on the scale should always run from the one end to the
other in the same direction so as to avoid negative readings. Fig. 1 5
Fig. 15.
represents the middle portion of a scale to be used with a mirror
and an inverting telescope.
This method of observation is the best when the motions are
slow. The observer sits at the telescope and sees the image of
the scale moving to right or to left past the vertical wire of the
telescope. With a clock beside him he can note the instant at
which a given division of the scale passes the wire, or the division
of the scale which is passing at a given tick of the clock, and he
can also record the extreme limits of each oscillation.
When the motion is more rapid it becomes impossible to read
the divisions of the scale except at the instants of rest at the
extremities of an oscillation. A conspicuous mark may be placed
at a known division of the scale, and the instant of transit of this
mark may be noted.
When the apparatus is very light, and the forces variable, the
motion is so prompt and swift that observation through a telescope
METHODS OF OBSERVATION. 93
would be useless. In this case the observer looks at the scale
directly, and observes the motions of the image of the vertical wire
thrown on the scale by a lamp.
It is manifest that since the image of the scale reflected by the
mirror and refracted by the object glass coincides with the vertical
wire, the image of the vertical wire, if sufficiently illuminated, will
coincide with the scale. To observe this the room is darkened, and
the concentrated rays of a lamp are thrown on the vertical wire
towards the object glass. A bright patch of light crossed by the
shadow of the wire is seen on the scale. Its motions can be
followed by the eye, and the division of the scale at which it comes
to rest can be fixed on by the eye and read off at leisure. If it be
desired to note the instant of the passage of the bright spot past a
given point on the scale, a pin or a bright metal wire may be
placed there so as to flash out at the time of passage.
By substituting a small hole in a diaphragm for the cross wire
the image becomes a small illuminated dot moving to right or left
on the scale, and by substituting for the scale a cylinder revolving
by clock work about a horizontal axis and covered with photo
graphic paper, the spot of light traces out a curve which can be
afterwards rendered visible. Each abscissa of this curve corresponds
to a particular time, and the ordinate indicates the angular
position of the mirror at that time. In this way an automatic
system of continuous registration of all the elements of terrestrial
magnetism has been established at Kew and other observatories.
In some cases the telescope is dispensed with, a vertical wire
is illuminated by a lamp placed behind it, and the mirror is a
concave one, which forms the image of the wire on the scale as
a dark line across a patch of light.
451.] In the Kew portable apparatus, the magnet is made in
the form of a tube, having at one end a lens, and at the other
a glass scale, so adjusted as to be at the principal focus of the lens.
Light is admitted from behind the scale, and after passing through
the lens it is viewed by means of a telescope.
Since the scale is at the principal focus of the lens, rays from
any division of the scale emerge from the lens parallel, and if
the telescope is adjusted for celestial objects, it will shew the scale
in optical coincidence with the cross wires of the telescope. If a
given division of the scale coincides with the intersection of the
cross wires, then the line joining that division with the optical
centre of the lens must be parallel to the line of collimation of
94 MAGNETIC MEASUKEMENTS. [45 2.
the telescope. By fixing the magnet and moving the telescope, we
may ascertain the angular value of the divisions of the scale, and
then, when the magnet is suspended and the position of the tele
scope known, we may determine the position of the magnet at
any instant by reading off the division of the scale which coincides
with the cross wires.
The telescope is supported on an arm which is centred in the
line of the suspension fibre, and the position of the telescope is
read off by verniers on the azimuth circle of the instrument.
This arrangement is suitable for a small portable magnetometer
in which the whole apparatus is supported on one tripod, and in
which the oscillations due to accidental disturbances rapidly
subside.
Determination of the Direction of the Axis of the Magnet, and of
the Direction of Terrestrial Magnetism.
452.] Let a system of axes be drawn in the magnet, of which the
axis of z is in the direction of the length of the bar, and x and y
perpendicular to the sides of the bar supposed a parallelepiped.
Let I, m, n and A, /u, v be the angles which the magnetic axis
and the line of collimation make with these axes respectively.
Let M be the magnetic moment of the magnet, let H be the
horizontal component of terrestrial magnetism, let Z be the vertical
component, and let 6 be the azimuth in which H acts, reckoned
from the north towards the west.
Let ( be the observed azimuth of the line of collimation, let
a be the azimuth of the stirrup, and (3 the reading of the index
of the torsion circle, then a — /3 is the azimuth of the lower end
of the suspension fibre.
Let y be the value of a — /3 when there is no torsion, then the
moment of the force of torsion tending to diminish a will be
T(a-/3-y),
where r is a coefficient of torsion depending on the nature of the
fibre.
To determine A, fix the stirrup so that y is vertical and up
wards, z to the north and so to the west, and observe the azimuth
f of the line of collimation. Then remove the magnet, turn it
through an angle TT about the axis of z and replace it in this
inverted position, and observe the azimuth f of the line of col
limation when y is downwards and x to the east,
452.] BISECTION OF MAGNETIC FOKCE. 95
f=a+f-A, (1)
r=a-|+A. (2)
Hence x = |+i(f-0. (3)
Next, hang the stirrup to the suspension fibre, and place the
magnet in it, adjusting it carefully so that y may be vertical and
upwards, then the moment of the force tending to increase a is
1— T (a— /3 — y). (4)
But if C is the observed azimuth of the line of collimation
C=a+|-A, (5)
so that the force may be written
MHsin *» sin (d - f + J- A) -T (f + A- - — 0 - y) • (6)
When the apparatus is in equilibrium this quantity is zero for
a particular value of f
When the apparatus never comes to rest, but must be observed
in a state of vibration, the value of £ corresponding to the position
of equilibrium may be calculated by a method which will be
described in Art. 735.
When the force of torsion is small compared with the moment
of the magnetic force, we may put d — £+ 1—\ for the sine of that
angle.
• If we give to /3, the reading of the torsion circle, two different
values, p! and /32, and if £ and £2 are the corresponding values of £
MHsinm^-Q = r (£-£_& + &), (7)
or, if we put
" , (8)
and equation (7) becomes, dividing by Jf/Jsin m,
-^-y = 0. (9)
If we now reverse the magnet so that y is downwards, and
adjust the apparatus till y is exactly vertical, and if f is the new
value of the azimuth, and 5' the corresponding declination,
/(f-X + -/3-y=0> (10)
whence - = i (f+C') + i/ (C+C'-2(/3-f y)). (11)
96 MAGNETIC MEASUREMENTS. [452.
The reading of the torsion circle should now be adjusted, so that
the coefficient of r may be as nearly as possible zero. For this
purpose we must determine y, the value of a — (3 when there is no
torsion. This may be done by placing a non-magnetic bar of the
same weight as the magnet in the stirrup, and determining a — /3
when there is equilibrium. Since / is small, great accuracy is not
required. Another method is to use a torsion bar of the same
weight as the magnet, containing within it a very small magnet
whose magnetic moment is - of that of the principal magnet.
Ifi
Since r remains the same, / will become m'} and if (^ and f/ are
the values of ( as found by the torsion bar,
6 = iCt + fiO+i*!" (£ + &'- 2 (/3 + y)). (12)
Subtracting this equation from (11),
2(»-l)(/3 + y) = (» + ^)(CI + C1')-(l + ^,)tf+O. (13)
Having found the value of /3-fy in this way, /3, the reading of
the torsion circle, should be altered till
f+f'-2(/3 + y) = 0, (14)
as nearly as possible in the ordinary position of the apparatus.
Then, since r' is a very small numerical quantity, and since its
coefficient is very small, the value of the second term in the ex
pression for 5 will not vary much for small errors in the values
of T and y, which are the quantities whose values are least ac
curately known.
The value of 8, the magnetic declination, may be found in this
way with considerable accuracy, provided it remains constant during
the experiments, so that we may assume 5'= 8.
When great accuracy is required it is necessary to take account
of the variations of 8 during the experiment. For this purpose
observations of another suspended magnet should be made at the
same instants that the different values of £ are observed, and if
r], if are the observed azimuths of the second magnet corresponding
to f and f ', and if 8 and 8' are the corresponding values of 8, then
8'-8 = rj'-r?. (15)
Hence, to find the value of 8 we must add to (11) a correction
i (')-•?')•
The declination at the time of the first observation is therefore
8 = 4(C+r+ ^-770 + 4/^+^-2/3-2^. (16)
453-] OBSERVATION OP DEFLEXION. 97
To find the direction of the magnetic axis within the magnet
subtract (10) from (9) and add (15),
^ = A + i(f-r)-H^-^Hi^(f-r-f2A-7r). (17)
By repeating the experiments with the bar on its two edges, so
that the axis of OB is vertically upwards and downwards, we can
find the value of m. If the axis of collimation is capable of ad
justment it ought to be made to coincide with the magnetic axis
as nearly as possible, so that the error arising from the magnet not
being exactly inverted may be as small as possible *.
On the Measurement of Magnetic Forces.
453.] The most important measurements of magnetic force are
those which determine M, the magnetic moment of a magnet,
and //, the intensity of the horizontal component of terrestrial
magnetism. This is generally done by combining the results of
two experiments, one of which determines the ratio and the other
the product of these two quantities.
The intensity of the magnetic force due to an infinitely small
magnet whose magnetic moment is M, at a point distant r from
the centre of the magnet in the positive direction of the axis of
the magnet, is ^ = 2— (I)
and is in the direction of r. If the magnet is of finite size but
spherical, and magnetized uniformly in the direction of its axis,
this value of the force will still be exact. If the magnet is a
solenoidal bar magnet of length 2 It,
*=2*(l + 2§ + sg + &c.). 00
If the magnet be of any kind, provided its dimensions are all
small compared with r,
JL)+fcc., (3)
where Alt A2, &c. are coefficients depending on the distribution of
the magnetization of the bar.
Let H be the intensity of the horizontal part of terrestrial
magnetism at any place. H is directed towards magnetic north.
Let r be measured towards magnetic west, then the magnetic force
at the extremity of r will be H towards the north and R towards
* See a Paper on 'Imperfect Inversion,' by W. Swan. Trans. R. S. Edin.,
vol. xxi (1855), p. 349.
VOL. TT. H
98 MAGNETIC MEASUREMENTS. [453-
the west. The resultant force will make an angle 0 with the
magnetic meridian, measured towards the west, and such that
(4)
Hence, to determine -~= we proceed as follows : —
JdL
The direction of the magnetic north having been ascertained, a
magnet, whose dimensions should not be too great, is suspended
as in the former experiments, and the deflecting magnet M is
placed so that its centre is at a distance r from that of the sus
pended magnet, in the same horizontal plane, and due magnetic
east.
The axis of M is carefully adjusted so as to be horizontal and
in the direction of r.
The suspended magnet is observed before M is brought near
and also after it is placed in position. If 0 is the observed deflexion,
we have, if we use the approximate formula ( 1 ),
f=^tau*; (5)
or, if we use the formula (3),
•.-•••. \ JrHan^l + ^i+^+fec. (6)
Here we must bear in mind that though the deflexion 0 can
be observed with great accuracy, the distance r between the centres
of the magnets is a quantity which cannot be precisely deter
mined, unless both magnets are fixed and their centres defined
by marks.
This difficulty is overcome thus :
The magnet M is placed on a divided scale which extends east
and west on both sides of the suspended magnet. The middle
point between the ends of M is reckoned the centre of the magnet.
This point may be marked on the magnet and its position observed
on the scale, or the positions of the ends may be observed and
the arithmetic mean taken. Call this Sj, and let the line of the
suspension fibre of the suspended magnet when produced cut the
scale at *0, then r1 = s1 — s0) where ^ is known accurately and s0 ap
proximately. Let 01 be the deflexion observed in this position of M.
Now reverse M, that is, place it on the scale with its ends
reversed, then ^ will be the same, but M and Alt A3, &c. will
have their signs changed, so that if 02 is ^ne deflexion,
- I r,»tan 9, = 1 -A, ± + J,± -&c. (7)
454-] DEFLEXION OBSERVATIONS. 99
Taking the arithmetical mean of (6) and (7),
i ^(tan^-tanfy = 1+^72 +^4^ + &c. (8)
Now remove M to the west side of the suspended magnet, and
place it with its centre at the point marked 2<$0 — s on the scale.
Let the deflexion when the axis is in the first position be 03, and
when it is in the second 04, then, as before,
2
Let us suppose that the true position of the centre of the sus
pended magnet is not SQ but <?0 -f or, then
(10)
and (V +, 2») = ,»(!. + 'l^ + &c.); (11)
O
and since -^ may be neglected if the measurements are .carefully
made, we are sure that we may take the arithmetical mean of rLn
and r2n for rn.
Hence, taking the arithmetical mean of (8) and (9),
--^
or, making
= 1 + A2~ +&c., (12)
- (tan Ol — tan 62 + tan 03 — tan 04) = D, (13)
454.] We may now regard D and r as capable of exact deter
mination.
The quantity A2 can in no case exceed 2^2, where L is half the
length of the magnet, so that when r is considerable compared
with L we may neglect the term in A2 and determine the ratio
of H to M at once. We cannot, however, assume that A2 is equal
to 2i/2, for it may be less, and may even be negative for a magnet
whose largest dimensions are transverse to the axis. The term
in A±, and all higher terms, may safely be neglected.
To eliminate A2, repeat the experiment, using distances rlt ra, ?*3,
&c., and let the values of D be J)19 D2, #3, &c., then
-2M(l , 4
2~~iT^ + ^
&c. &c.
II 2
100 MAGNETIC MEASUREMENTS. [454-
If we suppose that the probable errors of these equations are
equal, as they will be if they depend on the determination of D
only, and if there is no uncertainty about r, then, by multiplying
each equation by r~3 and adding the results, we obtain one equation,
and by multiplying each equation by r~5 and adding we obtain
another, according to the general rule in the theory of the com
bination of fallible measures when the probable error of each
equation is supposed the same.
Let us write
2(Vr-*) for AT3 + -02V3 + A^f 3 + &c.,
and use similar expressions for the sums of other groups of symbols,
then the two resultant equations may be written
*} = (2 (r-&) + 4 2
O TUT
2 (J)r~5) = -g- (2 (*-«) + A2 2
whence
1 W
-=- 2 /-6 2r-10~2/-82 = 2 Z>r
and 4>{2 (D?-3) 2 (r~10)-2 (Dr~5) 2 (*-8)}
= 2 (Dr-B) 2 (r-«)-2 (Dr~*) 2 (r-8).
The value of A2 derived from these equations ought to be less
than half the square of the length of the magnet M. If it is not
we may suspect some error in the observations. This method of
observation and reduction was given by Gauss in the ( First Report
of the Magnetic Association/
When the observer can make only two series of experiments at
2M
distances r± and r2, the value of -=- derived from these experi
ments is
- -
If 5Z)X and bD2 are the actual errors of the observed deflexions
^ and _Z)2, the actual error of the calculated result Q will be
If we suppose the errors 8^ and bD2 to be independent, and
that the probable value of either is SD, then the probable value
of the error in the calculated value of Q will be 5 Q, where
455-1 METHODS OF TANGENTS AND SINES. 101
If we suppose that one of these distances, say the sinaHar,; ijs-
given, the value of the greater distance may be determined so as
to make b Q a minimum. This condition leads to an equation of
the fifth degree in rf^ which has only one real root greater than
r22. From this the best value of ^ is found to be rx = 1.3189/2*.
If one observation only is taken the best distance is when
bD r-lr
-— = x/3 — ,
D ° r
where b D is the probable error of a measurement of deflexion, and
br is the probable error of a measurement of distance.
Method of Sines.
455.] The method which we have just described may be called
the Method of Tangents, because the tangent of the deflexion is
a measure of the magnetic force.
If the line rl5 instead of being measured east or west, is adjusted
till it is at right angles with the axis of the deflected magnet,
then R is the same as before, but in order that the suspended
magnet may remain perpendicular to r, the resolved part of the
force H in the direction of r must be equal and opposite to R.
Hence, if 0 is the deflexion, R — Hsm 0.
This method is called the Method of Sines. It can be applied
only when R is less than H.
In the Kew portable apparatus this method is employed. The
suspended magnet hangs from a part of the apparatus which
revolves along with the telescope and the arm for the deflecting
magnet, and the rotation of the whole is measured on the azimuth
circle.
The apparatus is first adjusted so that the axis of the telescope
coincides with the mean position of the line of collimation of the
magnet in its undisturbed state. If the magnet is vibrating, the
true azimuth of magnetic north is found by observing the ex
tremities of the oscillation of the transparent scale and making the
proper correction of the reading of the azimuth circle.
The deflecting magnet is then placed upon a straight rod which
passes through the axis of the revolving apparatus at right angles
to the axis of the telescope, and is adjusted so that the axis of the
deflecting magnet is in a line passing through the centre of the
suspended magnet.
The whole of the revolving apparatus is then moved till the line
* See Airy's Magnetism.
102 MAGNETIC MEASUREMENTS. [45$.
of coilimation of the suspended magnet again coincides with the
axis of the telescope, and the new azimuth reading is corrected,
if necessary, by the mean of the scale readings at the extremities
of an oscillation.
The difference of the corrected azimuths gives the deflexion, after
which we proceed as in the method of tangents, except that in the
expression for D we put sin & instead of tan 6.
In this method there is no correction for the torsion of the sus
pending fibre, since the relative position of the fibre, telescope,
and magnet is the same at every observation.
The axes of the two magnets remain always at right angles in
this method, so that the correction for length can be more ac
curately made.
456.] Having thus measured the ratio of the moment of the
deflecting magnet to the horizontal component of terrestrial mag
netism, we have next to find the product of these quantities, by
determining the moment of the couple with which terrestrial mag
netism tends to turn the same magnet when its axis is deflected
from the magnetic meridian.
There are two methods of making this measurement, the dy
namical, in which the time of vibration of the magnet under the
action of terrestrial magnetism is observed, and the statical, in
which the magnet is kept in equilibrium between a measurable
statical couple and the magnetic force.
The dynamical method requires simpler apparatus and is more
accurate for absolute measurements, but takes up a considerable
time, the statical method admits of almost instantaneous measure
ment, and is therefore useful in tracing the changes of the intensity
of the magnetic force, but it requires more delicate apparatus, and
is not so accurate for absolute measurement.
Method of Vibrations.
The magnet is suspended with its magnetic axis horizontal, and
is set in vibration in small arcs. The vibrations are observed by
means of any of the methods already described.
A point on the scale is chosen corresponding to the middle of
the arc of vibration. The instant of passage through this point
of the scale in the positive direction is observed. If there is suffi
cient time before the return of the magnet to the same point, the
instant of passage through the point in the negative direction is
also observed, and the process is continued till n+I positive and
456.] TIME OF VIBKATION. 103
n negative passages have been observed. If the vibrations are
too rapid to allow of every consecutive passage being observed,
every third or every fifth passage is observed, care being taken that
the observed passages are alternately positive and negative.
Let the observed times of passage be T1} T2, T2n+1, then if
we put I 4 + y + y 4 &c.
then Tn+1 is the mean time of the positive passages, and ought
to agree with T'n+v the mean time of the negative passages, if the
point has been properly chosen. The mean of these results is
to be taken as the mean time of the middle passage.
After a large number of vibrations have taken place, but before
the vibrations have ceased to be distinct and regular, the observer
makes another series of observations, from which he deduces the
mean time of the middle passage of the second series.
By calculating the period of vibration either from the first
series of observations or from the second, he ought to be able to
be certain of the number of whole vibrations which have taken
place in the interval between the time of middle passage in the two
series. Dividing the interval between the mean times of middle
passage in the two series by this number of vibrations, the mean
time of vibration is obtained.
The observed time of vibration is then to be reduced to the
time of vibration in infinitely small arcs by a formula of the same
kind as that used in pendulum observations, and if the vibrations
are found to diminish rapidly in amplitude, there is another cor
rection for resistance, see Art. 740. These corrections, however, are
very small when the magnet hangs by a fibre, and when the arc of
vibration is only a few degrees.
The equation of motion of the magnet is
- = 0
where 0 is the angle between the magnetic axis and the direction
of the force H, A is the moment of inertia of the magnet and
suspended apparatus, M is the magnetic moment of the magnet,
H the intensity of the horizontal magnetic force, and MHr' the
coefficient of torsion : / is determined as in Art. 452, and is a
very small quantity. The value of 0 for equilibrium is
T "y
00 = - - T 5 a very small angle,
104 MAGNETIC MEASUREMENTS. [457-
and the solution of the equation for small values of the amplitude,
C is f t \
0 = Ccos (2 TT -^ 4- a) + 00,
where T is the periodic time, and C the amplitude, and
yr2
whence we find the value of MH9
Here T is the time of a complete vibration determined from
observation. A, the moment of inertia, is found once for all for
the magnet, either by weighing and measuring it if it is of a
regular figure, or by a dynamical process of comparison with a body
whose moment of inertia is known.
Combining this value of Mil with that of -~ formerly obtained,
we get Jp
and //*
457.] We have supposed that //and M continue constant during
the two series of experiments. The fluctuations of // may be
ascertained by simultaneous observations of the bifilar magnet
ometer to be presently described, and if the magnet has been in
use for some time, and is not exposed during the experiments to
changes of temperature or to concussion, the part of M which de
pends on permanent magnetism may be assumed to be constant.
All steel magnets, however, are capable of induced magnetism
depending on the action of external magnetic force.
Now the magnet when employed in the deflexion experiments
is placed with its axis east and west, so that the action of ter
restrial magnetism is transverse to the magnet, and does not tend
to increase or diminish M. When the magnet is made to vibrate,
its axis is north and south, so that the action of terrestrial mag
netism tends to magnetize it in the direction of the axis, and
therefore to increase its magnetic moment by a quantity Jc //, where
k is a coefficient to be found by experiments on the magnet.
There are two ways in which this source of error may be avoided
without calculating Jc, the experiments being arranged so that the
magnet shall be in the same condition when employed in deflecting
another magnet and when itself swinging.
457-] ELIMINATION OF INDUCTION. 105
We may place the deflecting magnet with its axis pointing
north, at a distance r from the centre of the suspended magnet,
the line r making an angle whose cosine is \/J with the magnetic
meridian. The action of the deflecting magnet on the suspended
one is then at right angles to its own direction, and is equal to
Here M is the magnetic moment when the axis points north,
as in the experiment of vibration, so that no correction has to be
made for induction.
This method, however, is extremely difficult, owing to the large
errors which would be introduced by a slight displacement of the
deflecting magnet, and as the correction by reversing the deflecting
magnet is not applicable here, this method is not to be followed
except when the object is to determine the coefficient of induction.
The following method, in which the magnet while vibrating is
freed from the inductive action of terrestrial magnetism, is due to
Dr. J. P. Joule *.
Two magnets are prepared whose magnetic moments are as
nearly equal as possible. In the deflexion experiments these mag
nets are used separately, or they may be placed simultaneously
on opposite sides of the suspended magnet to produce a greater
deflexion. In these experiments the inductive force of terrestrial
magnetism is transverse to the axis.
Let one of these magnets be suspended, and let the other be
placed parallel to it with its centre exactly below that of the sus
pended magnet, and with its axis in the same direction. The force
which the fixed magnet exerts on the suspended one is in the
opposite direction from that of terrestrial magnetism. If the fixed
magnet be gradually brought nearer to the suspended one the time
of vibration will increase, till at a certain point the equilibrium will
cease to be stable, and beyond this point the suspended magnet
will make oscillations in the reverse position. By experimenting
in this way a position of the fixed magnet is found at which it
exactly neutralizes the effect of terrestrial magnetism on the sus
pended one. The two magnets are fastened together so as to be
parallel, with their axes turned the same way, and at the distance
just found by experiment. They are then suspended in the usual
way and made to vibrate together through small arcs.
* Proc. Phil. S., Manchester, March 19, 1867.
106 MAGNETIC MEASUREMENTS. [45 8.
The lower magnet exactly neutralizes the effect of terrestrial
magnetism on the upper one, and since the magnets are of equal
moment, the upper one neutralizes the inductive action of the earth
on the lower one.
The value of M is therefore the same in the experiment of
vibration as in the experiment of deflexion, and no correction for
induction is required.
458.] The most accurate method of ascertaining the intensity of
the horizontal magnetic force is that which we have just described.
The whole series of experiments, however, cannot be performed with
sufficient accuracy in much less than an hour, so that any changes
in the intensity which take place in periods of a few minutes would
escape observation. Hence a different method is required for ob
serving the intensity of the magnetic force at any instant.
The statical method consists in deflecting the magnet by means
of a statical couple acting in a horizontal plane. If L be the
moment of this couple, M the magnetic moment of the magnet,
// the horizontal component of terrestrial magnetism, and 0 the
deflexion, M H sin 0 = L.
Hence, if L is known in terms of 0, MH can be found.
The couple L may be generated in two ways, by the torsional
elasticity of a wire, as in the ordinary torsion balance, or by the
weight of the suspended apparatus, as in the bifilar suspension.
In the torsion balance the magnet is fastened to the end of a
vertical wire, the upper end of which can be turned round, and its
rotation measured by means of a torsion circle.
We have then
X, = r(a — a0 — 6) = Mil sin 6.
Here a0 is the value of the reading of the torsion circle when the
axis of the magnet coincides with the magnetic meridian, and a is
the actual reading. If the torsion circle is turned so as to bring
the magnet nearly perpendicular to the magnetic meridian, so that
e = ~tf, then r(a-a0- + 00
or
By observing 0', the deflexion of the magnet when in equilibrium,
we can calculate Mil provided we know r.
If we only wish to know the relative value of H at different
times it is not necessary to know either M or T.
We may easily determine T in absolute measure by suspending
459-] BIFILAB SUSPENSION. 107
a non-magnetic body from the same wire and observing its time
of oscillation, then if A is the moment of inertia of this body, and
T the time of a complete vibration,
The chief objection to the use of the torsion balance is that the
zero-reading a0 is liable to change. Under the constant twisting
force, arising from the tendency of the magnet to turn to the north,
the wire gradually acquires a permanent twist, so that it becomes
necessary to determine the zero-reading of the torsion circle afresh
at short intervals of time.
Bifilar Suspension.
459.] The method of suspending the magnet by two wires or
fibres was introduced by Gauss and Weber. As the bifilar sus
pension is used in many electrical instruments, we shall investigate
it more in detail. The general appearance of the suspension is
shewn in Fig. 16, and Fig. 17 represents the projection of the wires
on a horizontal plane.
AB and A'B' are the projections of the two wires.
AA and BB' are the lines joining the upper and the lower ends
of the wires.
a and b are the lengths of these lines.
a and /3 their azimuths.
TFand W the vertical components of the tensions of the wires.
Q and Q' their horizontal components.
h the vertical distance between AA and BB'.
The forces which act on the magnet are — its weight, the couple
arising from terrestrial magnetism, the torsion of the wires (if any)
and their tensions. Of these the effects of magnetism and of
torsion are of the nature of couples. Hence the resultant of the
tensions must consist of a vertical force, equal to the weight of the
magnet, together with a couple. The resultant of the vertical
components of the tensions is therefore along the line whose pro
jection is 0, the intersection of A A and BB', and either of these
lines is divided in 0 in the ratio of W to W.
The horizontal components of the tensions form a couple, and
are therefore equal in magnitude and parallel in direction. Calling
either of them Q, the moment of the couple which they form is
L=Q.PF, (1)
where PP7 is the distance between the parallel lines AB and AB'.
108
MAGNETIC MEASUREMENTS.
[459-
To find the value of L we have the equations of moments
Qh = W. AB = Jr. AK> (2)
and the geometrical equation
(AB + A'ff) PPf = ab sin (a- ft), ( 3)
whence we obtain,
ab WW
1=
W+ W
r, sin(a-/3).
Fig. 16.
Fig. 17.
(4)
If m is the mass of the suspended apparatus, and g the intensity
of gravity, w+ W' = mg. (5)
If we also write W— W — nmg> (6)
L — - (i—.n?)m.ff-jr sin (a— ft).
we find L — - (1 —nz}mff — sin (a — /3V (7)
The value of L is therefore a maximum with respect to n when n
459-] BIFILAR SUSPENSION. 109
is zero, that is, when the weight of the suspended mass is equally
borne by the two wires.
We may adjust the tensions of the wires to equality by observing1
the time of vibration, and making it a minimum, or we may obtain
a self-acting adjustment by attaching the ends of the wires, as
in Fig. 16, to a pulley, which turns on its axis till the tensions
are equal.
The distance of the upper ends of the suspension wires is re
gulated by means of two other pullies. The distance between the
lower ends of the wires is also capable of adjustment.
By this adjustment of the tension, the couple arising from the
tensions of the wires becomes
T I ab . .
L = - -j- mg sin (a— -/3).
The moment of the couple arising from the torsion of the wires
is of the form T (y—p\
where r is the sum of the coefficients of torsion of the wires.
The wires ought to be without torsion when a = ft, we may
then make y — a.
The moment of the couple arising from the horizontal magnetic
force is of the form
MS BIU (3 — 0),
where 8 is the magnetic declination, and 0 is the azimuth of the
axis of the magnet. We shall avoid the introduction of unnecessary
symbols without sacrificing generality if we assume that the axis of
the magnet is parallel to £JB', or that /3 = 0.
The equation of motion then becomes
4--j72= MHsw(b — 0} + - ^-^sin(a — 0) + r(a-0). (8)
There are three principal positions of this apparatus.
(1) When a is nearly equal to 8. If T^ is the time of a complete
oscillation in this position, then
47r2^ lab
-yrr- = l-fi>"ff+T + MH. (9)
(2) When a is nearly equal to 8 + 77. If T2 is the time of a
complete oscillation in this position, the north end of the magnet
being now turned towards the south,
1 ab
^-jrWff + T-MH. (10)
The quantity on the right-hand of this equation may be made
130 MAGNETIC MEASUREMENTS. [459.
as small as we please by diminishing a or £, but it must not be
made negative, or the equilibrium of the magnet will become un
stable. The magnet in this position forms an instrument by which
small variations in the direction of the magnetic force may be
rendered sensible.
For when 5—0 is nearly equal to TT, sin (8 — 0) is nearly equal to
6 — by and we find
(8-a). (11)
= a-
7
l ah 71* rr
- ~j-mg-\-T —MH
4 fl
By diminishing the denominator of the fraction in the last term
we may make the variation of 0 very large compared with that of 8.
We should notice that the coefficient of 8 in this expression is
negative, so that when the direction of the magnetic force turns
in one direction the magnet turns in the opposite direction.
(3) In the third position the upper part of the suspension-
apparatus is turned round till the axis of the magnet is nearly
perpendicular to the magnetic meridian.
If we make
0-8=|+0/, and a-6 = p-P, (12)
the equation of motion may be written
(/:J-0'). (13)
If there is equilibrium when //= EQ and 0'= 0,
= 0, (14)
and if H is the value of the horizontal force corresponding to a
small angle 0/, x ^
- -j- mg cos /3 -|- T \
'--~ -
In order that the magnet may be in stable equilibrium it is
necessary that the numerator of the fraction in the second member
should be positive, but the more nearly it approaches zero, the
more sensitive will be the instrument in indicating changes in the
value of the intensity of the horizontal component of terrestrial
magnetism.
The statical method of estimating the intensity of the force
depends upon the action of an instrument which of itself assumes
46 1. J DTP. Ill
different positions of equilibrium for different values of the force.
Hence, by means of a mirror attached to the magnet and throwing1
a spot of light upon a photographic surface moved by clockwork,
a curve may be traced, from which the intensity of the force at any
instant may be determined according to a scale, which we may for
the present consider an arbitrary one.
460.] In an observatory, where a continuous system of regis
tration of declination and intensity is kept up either by eye ob
servation or by the automatic photographic method, the absolute
values of the declination and of the intensity, as well as the position
and moment of the magnetic axis of a magnet, may be determined
to a greater degree of accuracy.
For the declinometer gives the declination at every instant affected
by a constant error, and the bifilar magnetometer gives the intensity
at every instant multiplied by a constant coefficient. In the ex
periments we substitute for b, 8 + 80 where 8' is the reading of
the declinometer at the given instant, and 80 is the unknown but
constant error, so that 8' + 80 is the true declination at that instant.
In like manner for H, we substitute CH' where IF is the reading*
' " O
of the magnetometer on its arbitrary scale, and C is an unknown
but constant multiplier which converts these readings into absolute
measure, so that CH' is the horizontal force at a given instant.
The experiments to determine the absolute values of the quan
tities must be conducted at a sufficient distance from the declino
meter and magnetometer, so that the different magnets may not
sensibly disturb each other. The time of every observation must
be noted and the corresponding values of 8' and H' inserted. The
equations are then to be treated so as to find 80, the constant error
of the declinometer, and C the coefficient to be applied to the
readings of the magnetometer. When these are found the readings
of both instruments may be expressed in absolute measure. The
absolute measurements, however, must be frequently repeated in
order to take account of changes which may occur in the magnetic
axis and magnetic moment of the magnets.
461.] The methods of determining the vertical component of the
terrestrial magnetic force have not been brought to the same degree
of precision. The vertical force must act on a magnet which turns
about a horizontal axis. Now a body which turns about a hori
zontal axis cannot be made so sensitive to the action of small forces
as a body which is suspended by a fibre and turns about a vertical
axis. Besides this, the weight of a magnet is so large compared
112 MAGNETIC MEASUREMENTS. [461.
with the magnetic force exerted upon it that a small displace
ment of the centre of inertia by unequal dilatation, &c. produces
a greater effect on the position of the magnet than a considerable
change of the magnetic force.
Hence the measurement of the vertical force, or the comparison
of the vertical and the horizontal forces, is the least perfect part
of the system of magnetic measurements.
The vertical part of the magnetic force is generally deduced from
the horizontal force by determining the direction of the total force.
If i be the angle which the total force makes with its horizontal
component, i is called the magnetic Dip or Inclination, and if H
is the horizontal force already found, then the vertical force is
//tan i, and the total force is H sec i.
The magnetic dip is found by means of the Dip Needle.
The theoretical dip-needle is a magnet with an axis which passes
through its centre of inertia perpendicular to the magnetic axis
of the needle. The ends of this axis are made in the form of
cylinders of small radius, the axes of which are coincident with the
line passing through the centre of inertia. These cylindrical ends
rest on two horizontal planes and are free to roll on them.
When the axis is placed magnetic east and west, the needle
is free to rotate in the plane of the magnetic meridian, and if the
instrument is in perfect adjustment, the magnetic axis will set itself
in the direction of the total magnetic force.
It is, however, practically impossible to adjust a dip-needle so
that its weight does not influence its position of equilibrium,
because its centre of inertia, even if originally in the line joining
the centres of the rolling sections of the cylindrical ends, will cease
to be in this line when the needle is imperceptibly bent or un
equally expanded. Besides, the determination of the true centre
of inertia of a magnet is a very difficult operation, owing to the
interference of the magnetic force with that of gravity.
Let us suppose one end of the needle and one end of the
pivot to be marked. Let a line, real or imaginary, be drawn on
the needle, which we shall call the Line of Collimation. The
position of this line is read off on a vertical circle. Let 6 be the
angle which this line makes with the radius to zero, which we shall
suppose to be horizontal. Let A. be the angle which the magnetic
axis makes with the line of collimation, so that when the needle
is in this position the line of collimation is inclined 0 + A. to the
horizontal.
461.] DIP CIRCLE. 11.3
Let p be the perpendicular from the centre of inertia on the plane
on which the axis rolls, then p will be a function of 6, whatever
be the shape of the rolling surfaces. If both the rolling sections
of the ends of the axis are circular,
p — c — #sin(0+a) (1)
where a is the distance of the centre of inertia from the line joining
the centres of the rolling sections, and a is the angle which this
line makes with the line of collimation.
If M is the magnetic moment, m the mass of the magnet, and
g the force of gravity, I the total magnetic force, and i the dip, then,
by the conservation of energy, when there is stable equilibrium,
MIcos(0 + \ — i) — mgjp (2)
must be a maximum with respect to 0, or
MIsm(0 + \-i)=-m<?d^> (3)
= —mg a cos (6 + a),
if the ends of the axis are cylindrical.
Also, if T be the time of vibration about the position of equi
librium,
: /,x
MI+ mga sin (6+ a) = -^-
where A is the moment of inertia of the needle about its axis of
rotation.
In determining the dip a reading is taken with the dip circle in
the magnetic meridian and with the graduation towards the west.
Let 61 be this reading, then we have
MIsm(01 + \—i) = —m(/acos(0l + a). (5)
The instrument is now turned about a vertical axis through 180°,
so that the graduation is to the east, and if 02 is the new reading,
MIsm(02 + X — v+i) ——mga cos (02 + a). (6)
Taking (6) from (5), and remembering that 6^ is nearly equal to
i, and 02 nearly equal to TT— i, and that X is a small angle, such
that mgaK may be neglected in comparison with MI,
MI(0l—02-{-7f—2i') =—2mgaco$icosa. (7)
Now take the magnet from its bearings and place it in the
deflexion apparatus, Art. 453, so as to indicate its own magnetic
moment by the deflexion of a suspended magnet, then
M=\r*HD (8)
where D is the tangent of the deflexion.
VOL. II. I
114 MAGNETIC MEASUREMENTS. [461.
Next, reverse the magnetism of the needle and determine its
new magnetic moment M', by observing a new deflexion, the tan
gent of which is D' ', M> = i ^ H1)^ (9)
whence MD' = M'D. ( 1 0)
Then place it on its bearings and take two readings, 03 and 04,
in which 03 is nearly ir + i, and 04 nearly —i,
3/'/' sin (03 + A' — 77 — i) = mgaco8(0B+a), (11)
M'l' sin (04 + A' + i) = m g a cos (04 + a), (1 2)
whence, as before,
M'I(93— 04 — 77 — 2i) — 2mgacosicosa, (13)
adding (8),
Jf/^-^ + Tr — 2z') + lT/(03 — 04— IT — 2 a) = 0, (14)
or J9(01-02 + 7r-2;) + .Z/(03-04-7r-2*) = 0, (15)
whence we find the dip
-e4-Tr) , .
where D and _Z/ are the tangents of the deflexions produced by the
needle in its first and second magnetizations respectively.
In taking observations with the dip circle the vertical axis is
carefully adjusted so that the plane bearings upon which the axis of
the magnet rests are horizontal in every azimuth. The magnet being
magnetized so that the end A dips, is placed with its axis on the
plane bearings, and observations are taken with the plane of the circle
in the magnetic meridian, and with the graduated side of the circle
east. Each end of the magnet is observed by means of reading
microscopes carried on an arm which moves concentric with the
dip circle. The cross wires of the microscope are made to coincide
with the image of a mark on the magnet, and the position of the
arm is then read off on the dip circle by means of a vernier.
We thus obtain an observation of the end A and another of the
end B when the graduations are east. It is necessary to observe
both ends in order to eliminate any error arising from the axle
of the magnet not being concentric with the dip circle.
The graduated side is then turned west, and two more observ
ations are made.
The magnet is then turned round so that the ends of the axle
are reversed, and four more observations are made looking at the
other side of the magnet.
463.] JOULE'S SUSPENSION. 115
The magnetization of the magnet is then reversed so that the
end B dips, the magnetic moment is ascertained, and eight ohserva-
tions are taken in this state, and the sixteen observations combined
to determine the true dip.
462.] It is found that in spite of the utmost care the dip, as thus
deduced from observations made with one dip circle, differs per
ceptibly from that deduced from observations with another dip
circle at the same place. Mr. Broun has pointed out the effect
due to ellipticity of the bearings of the axle, arid how to correct
it by taking observations with the magnet magnetized to different
strengths.
The principle of this method may be stated thus. We shall
suppose that the error of any one observation is a small quantity
not exceeding a degree. We shall also suppose that some unknown
but regular force acts upon the magnet, disturbing it from its
true position.
If L is the moment of this force, 00 the true dip, and 0 the
observed dip, then
L = Jf/sin(0-00), (17)
= MI(0-00), (18)
since 0 — 6$ is small.
It is evident that the greater M becomes the nearer does the
needle approach its proper position. Now let the operation of
taking the dip be performed twice, first with the magnetization
equal to Mlt the greatest that the needle is capable of, and next
with the magnetization equal to M~29 a much smaller value but
sufficient to make the readings distinct and the error still moderate.
Let 01 and 62 be the dips deduced from these two sets of observ
ations, and let L be the mean value of the unknown disturbing
force for the eight positions of each determination, which we shall
suppose the same for both determinations. Then
L = M1i(01-e0) = M2i(02-00). (19)
If we find that several experiments give nearly equal values for
L, then we may consider that 00 must be very nearly the true value
of the dip.
463.] Dr. Joule has recently constructed a new dip-circle, in
which the axis of the needle, instead of rolling on horizontal agate
planes, is slung on two filaments of silk or spider's thread, the ends
I 2
116
MAGNETIC MEASUREMENTS.
[463-
of the filaments being attached to the arms of a delicate balance.
The axis of the needle thus rolls on two loops of silk fibre, and
Dr. Joule finds that its freedom of motion is much greater than
when it rolls on agate planes.
In Fig. 18, NS is the needle, CC' is its axis, consisting of a
straight cylindrical wire, and PCQ, P'C'Q' are the filaments on which
the axis rolls. POQ is the
balance, consisting of a double
bent lever supported by a
wire, 0 0, stretched horizont
ally between the prongs of
a forked piece, and having
a counterpoise It which can
be screwed up or down, so
that the balance is in neutral
equilibrium about 0 0.
In order that the needle
may be in neutral equilibrium
as the needle rolls on the
filaments the centre of gra
vity must neither rise nor fall.
Hence the distance OC must
remain constant as the needle
rolls. This condition will be
fulfilled if the arms of the
balance OP and 0 Q are equal,
and if the filaments are at
right angles to the arms.
Dr. Joule finds that the
needle should not be more than
five inches long. When it is eight inches long, the bending of the
needle tends to diminish the apparent dip by a fraction of a minute.
The axis of the needle was originally of steel wire, straightened by
being brought to a red heat while stretched by a weight, but
Dr. Joule found that with the new suspension it is not necessary
to use steel wire, for platinum and even standard gold are hard
enough.
The balance is attached to a wire 00 about a foot long stretched
horizontally between the prongs of a fork. This fork is turned
round in azimuth by means of a circle at the top of a tripod which
supports the whole,. Six complete observations of the dip can be
464.]
VEETICAL FORCE.
117
obtained in one hour, and the average error of a single observation
is a fraction of a minute of arc.
It is proposed that the dip needle in the Cambridge Physical
Laboratory shall be observed by means of a double image instru
ment, consisting of two totally reflecting prisms placed as in
Fig. 19 and mounted on a vertical graduated circle, so that the
plane of reflexion may be turned round a horizontal axis nearly
coinciding with the prolongation of the axis of the suspended dip-
needle. The needle is viewed by means of a telescope placed
behind the prisms, and the two ends of the needle are seen together
as in Fig. 20. By turning the prisms about the axis of the vertical
circle, the images of two lines drawn on the needle may be made
to coincide. The inclination of the needle is thus determined from
the reading of the vertical circle.
Fig. 19.
Fig. 20.
The total intensity / of the magnetic force in the line of dip may
be deduced as follows from the times of vibration
in the four positions already described,
T13 Tz, jP3,
5JL _L JL J_l.
" 2M+2' I Zi2 + Tf "h T* h T* )
The values of M and M' must be found by the method of deflexion
and vibration formerly described, and A is the moment of inertia of
the magnet about its axle.
The observations with a magnet suspended by a fibre are so
much more accurate that it is usual to deduce the total force from
the horizontal force from the equation
/= H sec 6,
where / is the total force, H the horizontal force, and 0 the dip.
464.] The process of determining the dip being a tedious one, is
not suitable for determining the continuous variation of the magnetic
118 MAGNETIC MEASUREMENTS. [464.
force. The most convenient instrument for continuous observa
tions is the vertical force magnetometer, which is simply a magnet
balanced on knife edges so as to be in stable equilibrium with its
magnetic axis nearly horizontal.
If Z is the vertical component of the magnetic force, M the
magnetic moment, and 0 the small angle which the magnetic axis
makes with the horizon
HZ = mgacQ&(a.~6),
where m is the mass of the magnet, g the force of gravity, a the
distance of the centre of gravity from the axis of suspension, and
a the angle which the plane through the axis and the centre of
gravity makes with the magnetic axis.
Hence, for the small variation of vertical force bZ, there will be
a variation of the angular position of the magnet bO such that
In practice this instrument is not used to determine the absolute
value of the vertical force, but only to register its small variations.
For this purpose it is sufficient to know the absolute value of Z
when 0 = 0, and the value of -y-r •
civ
The value of Z, when the horizontal force and the dip are known,
is found from the equation Z = ZTtan00, where 00 is the dip and
H the horizontal force.
To find the deflexion due to a given variation of Z, take a magnet
and place it with its axis east and west, and with its centre at a
known distance i\ east or west from the declinometer, as in ex
periments on deflexion, and let the tangent of deflexion be Dl .
Then place it with its axis vertical and with its centre at a
distance rz above or below the centre of the vertical force mag
netometer, and let the tangent of the deflexion produced in the
magnetometer be D2. Then, if the moment of the deflecting
magnet is M, jr.
M^IIr^D^ = ^r^D2.
clZ r^ DL
Hence -7— = H -^ -~ •
dO r23 D2
The actual value of the vertical force at any instant is
7 7 +fidZ
& = &Q H- v -j^ >
where ZQ is the value of Z when Q = 0.
For continuous observations of the variations of magnetic force
464.] VERTICAL FOKCE. 119
at a fixed observatory the Unifilar Declinometer, the Bifilar Hori
zontal Force Magnetometer, and the Balance Vertical Force Mag
netometer are the most convenient instruments.
At several observatories photographic traces are now produced on
prepared paper moved by clock work, so that a continuous record
of the indications of the three instruments at every instant is formed.
These traces indicate the variation of the three rectangular com
ponents of the force from their standard values. The declinometer
gives the force towards mean magnetic west, the bifilar magnet
ometer gives the variation of the force towards magnetic north, and
the balance magnetometer gives the variation of the vertical force.
The standard values of these forces, or their values when these
instruments indicate their several zeros, are deduced by frequent
observations of the absolute declination, horizontal force, and dip.
CHAPTER VIII.
ON TERRESTRIAL MAGNETISM.
465.] OUR knowledge of Terrestrial Magnetism is derived from
the study of the distribution of magnetic force on the earth's sur
face at any one time, and of the changes in that distribution at
different times.
The magnetic force at any one place and time is known when
its three coordinates are known. These coordinates may be given
in the form of the declination or azimuth of the force, the dip
or inclination to the horizon, and the total intensity.
The most convenient method, however, for investigating the
general distribution of magnetic force on the earth's surface is to
consider the magnitudes of the three components of the force,
X=Hcosb, directed due north, \
Y=Hsmb, directed due west, (1)
Z = If tan 0, directed vertically downwards, )
where H denotes the horizontal force, 8 the declination, and 0
the dip.
If V is the magnetic potential at the earth's surface, and if we
consider the earth a sphere of radius a, then
Y i dr i dv dv , }
A = -- ^yj Y = - j — > ^=-^-7 (*)
a dl a cos I dK dr
where I is the latitude, and A. the longitude, and r the distance
from the centre of the earth.
A knowledge of V over the surface of the earth may be obtained
from the observations of horizontal force alone as follows.
Let FQ be the value of V at the true north pole, then, taking
the line-integral along any meridian, we find,
o, (3)
for the value of the potential on that meridian at latitude I.
466.] MAGNETIC SURVEY. 121
Thus the potential may be found for any point on the earth's
surface provided we know the value of X, the northerly component
at every point, and F0, the value of Fat the pole.
Since the forces depend not on the absolute value of V but
on its derivatives, it is not necessary to fix any particular value
for F0.
The value of V at any point may be ascertained if we know
the value of X along any given meridian, and also that of T over
the whole surface.
Let JF»»/j:tf+7*, W
where the integration is performed along the given meridian from
the pole to the parallel I, then
F= ^+«fVco8/#A, (5)
^AO
where the integration is performed along the parallel I from the
given meridian to the required point.
These methods imply that a complete magnetic survey of the
earth's surface has been made, so that the values of X or of Y
or of both are known for every point of the earth's surface at a
given epoch. What we actually know are the magnetic com
ponents at a certain number of stations. In the civilized parts of
the earth these stations are comparatively numerous ; in other places
there are large tracts of the earth's surface about which we have
no data.
Magnetic Survey.
466.] Let us suppose that in a country of moderate size, whose
greatest dimensions are a few hundred miles, observations of the
declination and the horizontal force have been taken at a con
siderable number of stations distributed fairly over the country.
Within this district we may suppose the value of V to be re
presented with sufficient accuracy by the formula
F= Vt + a(AJ + Ai\+\BJ*+EJ\+\3iK* + ^ (6)
whence X = A1 + Bl I + £2 X, (7)
Ycosl = A2 + £2l + 33\. (8)
Let there be n stations whose latitudes are ll} £2, ...&c. and
longitudes \lt A2, &c., and let X and 7 be found for each station.
Let J =
122 TERRESTRIAL MAGNETISM. [466-
/0 and A0 may be called the latitude and longitude of the central
station. Let
X0=-i(i)- and rocosJ0=:-2(rcosJ), (10)
tl ti
then X0 and Y0 are the values of X and Y at the imaginary central
station, then
\-\0), (11)
A-A0). (12)
We have n equations of the form of (11) and n of the form (12).
If we denote the probable error in the determination of X by £,
and that of Ycos I by q, then we may calculate f and r/ on
the supposition that they arise from errors of observation of H
and 8.
Let the probable error of H be ^, and that of 8, d, then since
dX — cos 5 . dff—Hsm 8 . db,
£2 = 7,2 COS2 8 + d*H* sin2 8
Similarly 7?2 = /fc2 sin2 8 + d2 //2 cos2 8.
If the variations of X and T from their values as given by equa
tions of the form (11) and (12) considerably exceed the probable
errors of observation, we may conclude that they are due to local
attractions, and then we have no reason to give the ratio of £ to r\
any other value than unity.
According to the method of least squares we multiply the equa
tions of the form (11) by r/, and those of the form (12) by £ to
make their probable error the same. We then multiply each
equation by the coefficient of one of the unknown quantities J3lt
H2, or BZ and add the results, thus obtaining three equations from
which to find B B and B.
in which we write for conciseness,
*1 = 2(^2)-»^ ^ =
Pl = 2(lX)-nlQXQ,
By calculating £19 J52, and J53, and substituting in equations
(11) and (12), we can obtain the values of X and Y at any point
within the limits of the survey free from the local disturbances
468.] MAGNETIC FEATURES OF THE EARTH. 123
which are found to exist where the rock near the station is magnetic,
as most igneous rocks are.
Surveys of this kind can be made only in countries where mag
netic instruments can be carried about and set up in a great many
stations. For other parts of the world we must be content to find
the distribution of the magnetic elements by interpolation between
their values at a few stations at great distances from each other.
467.] Let us now suppose that by processes of this kind, or
by the equivalent graphical process of constructing charts of the
lines of equal values of the magnetic elements, the values of X and
Y, and thence of the potential V, are known over the whole surface
of the globe. The next step is to expand V in the form of a series
of spherical surface harmonics.
If the earth were magnetized uniformly and in the same direction
throughout its interior, V would be an harmonic of the first degree,
the magnetic meridians would be great circles passing through two
magnetic poles diametrically opposite, the magnetic equator would
be a great circle, the horizontal force would be equal at all points
of the magnetic equator, and if H0 is this constant value, the value
at any other point would be H= //Ocos I', where V is the magnetic
latitude. The vertical force at any point would be Z = 2 HQ sin I' ,
and if Q is the dip, tan 6 = 2 tan I' .
In the case of the earth, the magnetic equator is defined to be
the line of no dip. It is not a great circle of the sphere.
The magnetic poles are defined to be the points where there is
no horizontal force or where the dip is 90°. There are two such
points, one in the northern and one in the southern regions, but
they are not diametrically opposite, and the line joining them is
not parallel to the magnetic axis of the earth.
468.] The magnetic poles are the points where the value of V
on the surface of the earth is a maximum or minimum, or is
stationary.
At any point where the potential is a minimum the north end
of the dip-needle points vertically downwards, and if a compass-
needle be placed anywhere near such a point, the north end will
point towards that point.
At points where the potential is a maximum the south end of
the dip-needle points downwards, and the south end of the compass-
needle points towards the point.
If there are p minima of V on the earth's surface there must be
p — \ other points, where the north end of the dip-needle points
124: TERRESTRIAL MAGNETISM. [469.
downwards, but where the compass-needle, when carried in a circle
round the point, instead of revolving so that its north end points
constantly to the centre, revolves in the opposite direction, so as to
turn sometimes its north end and sometimes its south end towards
the point.
If we call the points where the potential is a minimum true
north poles, then these other points may be called false north poles,
because the compass-needle is not true to them. If there are p
true north poles, there must be p — I false north poles, and in like
manner, if there are q true south poles, there must be y — 1 false
south poles. The number of poles of the same name must be odd,
so that the opinion at one time prevalent, that there are two north
poles and two south poles, is erroneous. According to Gauss there
is in fact only one true north pole and one true south pole on
the earth's surface, and therefore there are no false poles. The line
joining these poles is not a diameter of the earth, and it is not
parallel to the earth's magnetic axis.
469.] Most of the early investigators into the nature of the
earth's magnetism endeavoured to express it as the result of the
action of one or more bar magnets, the position of the poles of
which were to be determined. Gauss was the first to express the
distribution of the earth's magnetism in a perfectly general way by
expanding its potential in a series of solid harmonics, the coefficients
of which he determined for the first four degrees. These coeffi
cients are 24 in number, 3 for the first degree, 5 for the second,
7 for the third, and 9 for the fourth. All these terms are found
necessary in order to give a tolerably accurate representation of
the actual state of the earth's magnetism.
To find what Part of the Observed Magnetic Force is due to External
and what to Internal Causes.
470.] Let us now suppose that we have obtained an expansion
of the magnetic potential of the earth in spherical harmonics,
consistent with the actual direction and magnitude of the hori
zontal force at every point on the earth's surface, then Gauss has
shewn how to determine, from the observed vertical force, "whether
the magnetic forces are due to causes, such as magnetization or
electric currents, within the earth's surface, or whether any part
is directly due to causes exterior to the earth's surface.
Let V be the actual potential expanded in a double series of
spherical harmonics,
472.] SUBTERRANEAN OH CELESTIAL I 125
-2
The first series represents the part of the potential due to causes
exterior to the earth,, and the second series represents the part due
to causes within the earth.
The observations of horizontal force give us the sum of these
series when r — a, the radius of the earth. The term of the order i is
The observations of vertical force give us
Z=* — >
dr '
and the term of the order i in aZ is
Hence the part due to external causes is
and the part due to causes within the earth is
_ r-
The expansion of V has hitherto been calculated only for the
mean value of V at or near certain epochs. No appreciable part
of this mean value appears to be due to causes external to the
earth.
471.] We do not yet know enough of the form of the expansion
of the solar and lunar parts of the variations of V to determine
by tills method whether any part of these variations arises from
magnetic force acting from without. It is certain, however, as
the calculations of MM. Stoney and Chambers have shewn, that
the principal part of these variations cannot arise from any direct
magnetic action of the sun or moon, supposing these bodies to be
magnetic *.
472.] The principal changes in the magnetic force to which
attention has been directed are as follows.
* Professor Hornstein of Prague has discovered a periodic change in the magnetic
elements, the period of which is 26.33 days, almost exactly equal to that of the
synodic revolution of the sun, as deduced from the observation of sun-spots near his
equator. This method of discovering the time of rotation of the unseen solid body of
the sun by its effects on the magnetic needle is the first instalment of the repayment
by Magnetism of its debt to Astronomy. Akad., Wien, June 1,5, 1871. See Proc.
R.8., Nov. 16,1871.
126 TERRESTRIAL MAGNETISM. [473-
I. The more Regular Variations.
(1) The Solar variations, depending on the hour of the day and
the time of the year.
(2) The Lunar variations, depending on the moon's hour angle
and on her other elements of position.
(3) These variations do not repeat themselves in different years,,
but seem to be subject to a variation of longer period of about
eleven years.
(4) Besides this, there is a secular alteration in the state of the
earth's magnetism, which has been going on ever since magnetic
observations have been made, and is producing changes of the
magnetic elements of far greater magnitude than any of the varia
tions of small period.
II. The Disturbances.
473.] Besides the more regular changes, the magnetic elements
are subject to sudden disturbances of greater or less amount. It
is found that these disturbances are more powerful and frequent
at one time than at another, and that at times of great disturbance
the laws of the regular variations are masked, though they are very
distinct at times of small disturbance. Hence great attention has
been paid to these disturbances, and it has been found that dis
turbances of a particular kind are more likely to occur at certain
times of the day, and at certain seasons and intervals of time,
though each individual disturbance appears quite irregular. Besides
these more ordinary disturbances, there are occasionally times of
excessive disturbance, in which the magnetism is strongly disturbed
for a day or two. These are called Magnetic Storms. Individual
disturbances have been sometimes observed at the same instant
in stations widely distant.
Mr. Airy has found that a large proportion of the disturbances
at Greenwich correspond with the electric currents collected by
electrodes placed in the earth in the neighbourhood, and are such
as would be directly produced in the magnet if the earth-current,
retaining its actual direction, were conducted through a wire placed
underneath the magnet.
It has been found that there is an epoch of maximum disturbance
every eleven years, and that this appears to coincide with the epoch
of maximum number of spots in the sun.
474.] The field of investigation into which we are introduced
474-] VARIATIONS AND DISTURBANCES. 127
by the study of terrestrial magnetism is as profound as it is ex
tensive,
We know that the sun and moon act on the earth's magnetism.
It has been proved that this action cannot be explained by sup
posing these bodies magnets. The action is therefore indirect. In
the case of the sun part of it may be thermal action, but in the
case of the moon we cannot attribute it to this cause. Is it pos
sible that the attraction of these bodies, by causing strains in the
interior of the earth, produces (Art. 447) changes in the magnetism
already existing in the earth, and so by a kind of tidal action causes
the semidiurnal variations ?
But the amount of all these changes is very small compared with
the great secular changes of the earth's magnetism.
What cause, whether exterior to the earth or in its inner depth s,
produces such enormous changes in the earth's magnetism, that its
magnetic poles move slowly from one part of the globe to another ?
When we consider that the intensity of the magnetization of the
great globe of the earth is quite comparable with that which we
produce with much difficulty in our steel magnets, these immense
changes in so large a body force us to conclude that we are not yet
acquainted with one of the most powerful agents in nature,, the
scene of whose activity lies in those inner depths of the earth, to
the knowledge of which we have so few means of access.
PART IV.
ELECTROMAGNETISM.
CHAPTEK I.
ELECTROMAGNETIC FORCE.
475.] IT had been noticed by many different observers that in
certain cases magnetism is produced or destroyed in needles by
electric discharges through them or near them, and conjectures
of various kinds had been made as to the relation between mag
netism and electricity, but the laws of these phenomena, and the
form of these relations, remained entirely unknown till Hans
Christian Orsted *, at a private lecture to a few advanced students
at Copenhagen, observed that a wire connecting the ends of a
voltaic battery affected a magnet in its vicinity. This discovery
he published in a tract entitled Experiments circa effectum Conflictus
Electrici in Acum Magneticam, dated July 21, 1820.
Experiments on the relation of the magnet to bodies charged
with electricity had been tried without any result till Orsted
endeavoured to ascertain the effect of a wire heated by an electric
current. He discovered, however, that the current itself, and not
the heat of the wire, was the cause of the action, and that the
e electric conflict acts in a revolving manner,' that is, that a magnet
placed near a wire transmitting an electric current tends to set
itself perpendicular to the wire, and with the same end always
pointing forwards as the magnet is moved round the wire.
476.] It appears therefore that in the space surrounding a wire
* See another account of Orsted's discovery in a letter from Professor Hansteen in
the Life of Faraday by Dr. Bence Jones, vol. ii. p. 395.
478.]
STRAIGHT CURRENT.
129
transmitting an electric current a magnet is acted on by forces
depending on the position of the wire and on the strength of the
current. The space in which these forces act may therefore be
considered as a magnetic field, and we may study it in the same
way as we have already studied the field in the neighbourhood of
ordinary magnets, by tracing the course of the lines of magnetic
force, and measuring the intensity of the force at every point.
477.] Let us begin with the case of an indefinitely long straight
wire carrying an electric current. If a man were to place himself
in imagination in the position of the wire, so that the current
should flow from his head to his feet, then a magnet suspended
freely before him would set itself so that the end which points north
would, under the action of the current, point to his right hand.
The lines of magnetic force are everywhere at right angles to
planes drawn through the wire, and are there
fore circles each in a plane perpendicular to
the wire, which passes through its centre.
The pole of a magnet which points north, if
carried round one of these circles from left to
right, would experience a force acting always
in the direction of its motion. The other
pole of the same magnet would experience
a force in the opposite direction.
478.] To compare these forces let the wire
be supposed vertical, and the current a de
scending one, and let a magnet be placed on
an apparatus which is free to rotate about a
vertical axis coinciding with the wire. It
is found that under these circumstances the
current has no effect in causing the rotation
of the apparatus as a whole about itself as an axis. Hence the
action of the vertical current on the two poles of the magnet is
such that the statical moments of the two forces about the current
as an axis are equal and opposite. Let % and m2 be the strengths
of the two poles, rl and r2 their distances from the axis of the wire,
5\ and T2 the intensities of the magnetic force due to the current at
Fig. 21.
the two poles respectively, then the force on m1 is
and
s
since it is at right angles to the axis its moment l
Similarly that of the force on the other pole is m2T2r2, and since
there is no motion observed,
mlT1rl + m2T2r2 = 0.
VOL. II. K
130 ELECTROMAGNETIC FORCE. [479-
But we know that in all magnets
m-L + m^ = 0.
Hence T^ = T2r2,
or the electro magnetic force due to a straight current of infinite
length is perpendicular to the current, and varies inversely as the
distance from it.
479.] Since the product Tr depends on the strength of the
current it may be employed as a measure of the current. This
method of measurement is different from that founded upon elec
trostatic phenomena, and as it depends on the magnetic phenomena
produced by electric currents it is called the Electromagnetic system
of measurement. In the electromagnetic system if i is the current,
Tr = 2i.
480.] If the wire be taken for the axis of z} then the rectangular
components of T are
Here Xdx+Ydy+Zdz is a complete differential, being that of
Hence the magnetic force in the field can be deduced from a
potential function, as in several former instances, but the potential
is in this case a function having an infinite series of values whose
common difference is 4:iri. The differential coefficients of the
potential with respect to the coordinates have, however, definite and
single values at every point.
The existence of a potential function in the field near an electric
current is not a self-evident result of the principle of the con
servation of energy, for in all actual currents there is a continual
expenditure of the electric energy of the battery in overcoming the
resistance of the wire, so that unless the amount of this expenditure
were accurately known, it might be suspected that part of the
energy of the battery may be employed in causing work to be
done on a magnet moving in a cycle. In fact, if a magnetic pole,
m, moves round a closed curve which embraces the wire, work
is actually done to the amount of 4 TT m i. It is only for closed
paths which do not embrace the wire that the line-integral of the
force vanishes. We must therefore for the present consider the
law of force and the existence of a potential as resting on the
evidence of the experiment already described.
483.] MAGNETIC POTENTIAL. 131
481.] If we consider the space surrounding an infinite straight
line we shall see that it is a cyclic space, because it returns into
itself. If we now conceive a plane, or any other surface, com
mencing at the straight line and extending on one side of it
to infinity, this surface may be regarded as a diaphragm which
reduces the cyclic space to an acyclic one. If from any fixed point
lines be drawn to any other point without cutting the diaphragm,
and the potential be defined as the line-integral of the force taken
along one of these lines, the potential at any point will then have
a single definite value.
The magnetic field is now identical in all respects with that due
to a magnetic shell coinciding with this surface, the strength of
the shell being i. This shell is bounded on one edge by the infinite
straight line. Tho other parts of its boundary are at an infinite
distance from the part of the field under consideration.
482.] In all actual experiments the current forms a closed circuit
of finite dimensions. We shall therefore compare the magnetic
action of a finite circuit with that of a magnetic shell of which the
circuit is the bounding edge.
It has been shewn by numerous experiments, of which the
earliest are those of Ampere, and the most accurate those of Weber,
that the magnetic action of a small plane circuit at distances which
are great compared with the dimensions of the circuit is the same
as that of a magnet whose axis is normal to the plane of the circuit,
and whose magnetic moment is equal to the area of the circuit
multiplied by the strength of the current.
If the circuit be supposed to be filled up by a surface bounded
by the circuit and thus forming a diaphragm, and if a magnetic
shell of strength i coinciding with this surface be substituted for
the electric current, then the magnetic action of the shell on all
distant points will be identical with that of the current.
483.] Hitherto we have supposed the dimensions of the circuit
to be small compared with the distance of any part of it from
the part of the field examined. We shall now suppose the circuit
to be of any form and size whatever, and examine its action at any
point P not in the conducting wire itself. The following method,
which has important geometrical applications, was introduced by
Ampere for this purpose.
Conceive any surface S bounded by the circuit and not passing
through the point P. On this surface draw two series of lines
crossing each other so as to divide it into elementary portions, the
K 2
132 ELECTROMAGNETIC FORCE. [484.
dimensions of which are small compared with their distance from
P, and with the radii of curvature of the surface.
Round each of these elements conceive a current of strength i
to flow, the direction of circulation being the same in all the ele
ments as it is in the original circuit.
Along every line forming the division between two contiguous
elements two equal currents of strength i flow in opposite direc
tions.
The effect of two equal and opposite currents in the same place
is absolutely zero, in whatever aspect we consider the currents.
Hence their magnetic effect is zero. The only portions of the
elementary circuits which are not neutralized in this way are those
which coincide with the original circuit. The total effect of the
elementary circuits is therefore equivalent to that of the original
circuit.
484.] Now since each of the elementary circuits may be con
sidered as a small plane circuit whose distance from P is great
compared with its dimensions, we may substitute for it an ele
mentary magnetic shell of strength i whose bounding edge coincides
with the elementary circuit. The magnetic effect of the elementary
shell on P is equivalent to that of the elementary circuit. The
whole of the elementary shells constitute a magnetic shell of
strength i, coinciding with the surface 8 and bounded by the
original circuit, and the magnetic action of the whole shell on P
is equivalent to that of the circuit.
It is manifest that the action of the circuit is independent
of the form of the surface S9 which was drawn in a perfectly
arbitrary manner so as to fill it up. We see from this that the
action of a magnetic shell depends only on the form of its edge
and not on the form of the shell itself. This result we obtained
before, at Art. 410, but it is instructive to see how it may be
deduced from electromagnetic considerations.
The magnetic force due to the circuit at any point is therefore
identical in magnitude and direction with that due to a magnetic
shell bounded by the circuit and not passing through the point,
the strength of the shell being numerically equal to that of the
current. The direction of the current in the circuit is related to
the direction of magnetization of the shell, so that if a man were to
stand with his feet on that side of the shell which we call the
positive side, and which tends to point to the north, the current in
front of him would be from right to left.
486.] MAGNETIC POTENTIAL DUE TO A CIRCUIT. 133
485.] The magnetic potential of the circuit, however, differs
from that of the magnetic shell for those points which are in the
substance of the magnetic shell.
If co is the solid angle subtended at the point P by the magnetic
shell, reckoned positive when the positive or austral side of the shell
is next to P, then the magnetic potential at any point not in the
shell itself is coc/>, where $ is the strength of the shell. At any
point in the substance of the shell itself we may suppose the shell
divided into two parts whose strengths are ^ and c/>2, where
</>! -f c/>2 = c/>, such that the point is on the positive side of c^1 and
on the negative side of c/>2 . The potential at this point is
On the negative side of the shell the potential becomes $ (co— • 47r).
In this case therefore the potential is continuous, and at every
point has a single determinate value. In the case of the electric
circuit, on the other hand, the magnetic potential at every point
not in the conducting wire itself is equal to ia>, where i is the
strength of the current, and co is the solid angle subtended by the
circuit at the point, and is reckoned positive when the current, as
seen from P, circulates in the direction opposite to that of the hands
of a watch.
The quantity ^co is a function having an infinite series of values
whose common difference is 4 TT i. The differential coefficients of
id) with respect to the coordinates have, however, single and de
terminate values for every point of space.
486.] If a long thin flexible solenoidal magnet were placed in
the neighbourhood of an electric circuit, the north and south ends
of the solenoid would tend to move in opposite directions round
the wire, and if they were free to obey the magnetic force the
magnet would finally become wound round the wire in a close
coil. If it were possible to obtain a magnet having only one pole,
or poles of unequal strength, such a magnet would be moved round
and round the wire continually in one direction, but since the poles
of every magnet are equal and opposite, this result can never occur.
Faraday, however, has shewn how to produce the continuous rotation
of one pole of a magnet round an electric current by making it
possible for one pole to go round and round the current while
the other pole does not. That this process may be repeated in
definitely, the body of the magnet must be transferred from one
side of the current to the other once in each revolution. To do
this without interrupting the flow of electricity, the current is split
134 ELECTROMAGNETIC FORCE.
into two branches, so that when one branch is opened to let the
magnet pass the current continues to flow through the other.
Faraday used for this purpose a circular trough of mercury, as
shewn in Fig. 23, Art. 491. The current enters the trough through
the wire AB, it is divided at B, and after flowing through the arcs
£QP and BRP it unites at P, and leaves the trough through the
wire PO, the cup of mercury 0, and a vertical wire beneath 0,
down which the current flows.
The magnet (not shewn in the figure) is mounted so as to be
capable of revolving about a vertical axis through 0, and the wire
OP revolves with it. The body of the magnet passes through the
aperture of the trough, one pole, say the north pole, being beneath
the plane of the trough, and the other above it. As the magnet
and the wire OP revolve about the vertical axis, the current is
gradually transferred from the branch of the trough which lies in
front of the magnet to that which lies behind it, so that in every
complete revolution the magnet passes from one side of the current
to the other. The north pole of the magnet revolves about the
descending current in the direction N.E.S.W. and if w, o>' are the
solid angles (irrespective of sign) subtended by the circular trough
at the two poles, the work done by the electromagnetic force in a
complete revolution is
mi (ITT — o> — a/),
where m is the strength of either pole, and i the strength of the
current.
487.] Let us now endeavour to form a notion of the state of the
magnetic field near a linear electric circuit.
Let the value of o>, the solid angle subtended by the circuit,
be found for every point of space, and let the surfaces for which
co is constant be described. These surfaces will be the equipotential
surfaces. Each of these surfaces will be bounded by the circuit,
and any two surfaces, o^ and o>2, will meet in the circuit at an
angle i(o>1-<i)2).
Figure XVIII, at the end of this volume, represents a section
of the equipotential surfaces due to a circular current. The small
circle represents a section of the conducting wire, and the hori
zontal line at the bottom of the figure is the perpendicular to the
plane of the circular current through its centre. The equipotential
surfaces, 24 of which are drawn corresponding to a series of values
of CD differing by — > are surfaces of revolution, having this line for
489.] ACTION OF A CIRCUIT ON A MAGNETIC SYSTEM. 135
their common axis. They are evidently oblate figures, being flat
tened in the direction of the axis. They meet each other in the line
of the circuit at angles of 1 5°.
The force acting on a magnetic pole placed at any point of an
equipotential surface is perpendicular to this surface, and varies
inversely as the distance between consecutive surfaces. The closed
curves surrounding the section of the wire in Fig. XVIII are the
lines of force. They are copied from Sir W. Thomson's Paper on
'Vortex Motion*.' See also Art. 702.
Action of an Electric Circuit on any Magnetic System.
488.] We are now able to deduce the action of an electric circuit
on any magnetic system in its neighbourhood from the theory of
magnetic shells. For if we construct a magnetic shell, whose
strength is numerically equal to the strength of the current, and
whose edge coincides in position with the circuit, while the shell
itself does not pass through any part of the magnetic system, the
action of the shell on the magnetic system will be identical with
that of the electric circuit.
Reaction of the Magnetic System on the Electric Circuit.
489.] From this, applying the principle that action and reaction
are equal and opposite, we conclude that the mechanical action of
the magnetic system on the electric circuit is identical with its
action on a magnetic shell having the circuit for its edge.
The potential energy of a magnetic shell of strength $ placed
in a field of magnetic force of which the potential is T, is, by
Art. 410,
T- -J- >
x dy dz'
where I, m, n are the direction-cosines of the normal drawn from the
positive side of the element dS of the shell, and the integration
is extended over the surface of the shell.
Now the surface-integral
where #, I, c are the components of the magnetic induction, re
presents the quantity of magnetic induction through the shell, or,
* Trans. R. 8. Edin., vol. xxv. p. 217, (1869).
136 ELECTROMAGNETIC FORCE. [490.
in the language of Faraday, the number of lines of magnetic in
duction, reckoned algebraically, which pass through the shell from
the negative to the positive side, lines which pass through the
shell in the opposite direction being reckoned negative.
Remembering that the shell does not belong to the magnetic
system to which the potential V is due, and that the magnetic
force is therefore equal to the magnetic induction, we have
dV dV dV
a= -- =-, b= -- =-, c = -- j->
dx dy dz
and we may write the value of M,
M=-<t>N.
If bx1 represents any displacement of the shell, and X1 the force
acting on the shell so as to aid the displacement, then by the
principle of conservation of energy,
"= 0,
^
or X = 6 --
r §x
We have now determined the nature of the force which cor
responds to any given displacement of the shell. It aids or resists
that displacement accordingly as the displacement increases or
diminishes N, the number of lines of induction which pass through
the shell.
The same is true of the equivalent electric circuit. Any dis
placement of the circuit will be aided or resisted accordingly as it
increases or diminishes the number of lines of induction which pass
through the circuit in the positive direction.
We must remember that the positive direction of a line of
magnetic induction is the direction in which the pole of a magnet
which points north tends to move along the line, and that a line
of induction passes through the circuit in the positive direction,
when the direction of the line of induction is related to the
direction of the current of vitreous electricity in the circuit as
the longitudinal to the rotational motion of a right-handed screw.
See Art. 23.
490.] It is manifest that the force corresponding to any dis
placement of the circuit as a whole may be deduced at once from
the theory of the magnetic shell. But this is not all. If a portion
of the circuit is flexible, so that it may be displaced independently
of the rest, we may make the edge of the shell capable of the same
kind of displacement by cutting up the surface of the shell into
49O.] FOKCE ACTING ON A CUKRENT. 137
a sufficient number of portions connected by flexible joints. Hence
we conclude that if by the displacement of any portion of the circuit
in a given direction the number of lines of induction which pass
through the circuit can be increased, this displacement will be aided
by the electromagnetic force acting on the circuit.
Every portion of the circuit therefore is acted on by a force
urging it across the lines of magnetic induction so as to include
a greater number of these lines within the embrace of the circuit,
and the work done by the force during this displacement is
numerically equal to the number of the additional lines of in
duction multiplied by the strength of the current.
Let the element ds of a circuit, in which a current of strength
i is flowing, be moved parallel to itself through a space §x, it will
sweep out an area in the form of a parallelogram whose sides are
parallel and equal to ds and bx respectively.
If the magnetic induction is denoted by 33, and if its direction
makes an angle e with the normal to the parallelogram, the value
of the increment of N corresponding to the displacement is found
by multiplying the area of the parallelogram by 33 cos e. The result
of this operation is represented geometrically by the volume of a
parallelepiped whose edges represent in magnitude and direction
8ar, ds, and 33, and it is to be reckoned positive if when we point
in these three directions in the order here given the pointer
moves round the diagonal of the parallelepiped in the direction of
the hands of a watch. The volume of this parallelepiped is equal
to Xb%.
If 0 is the angle between ds and 33, the area of the parallelogram
is ds . 33 sin 6, and if 77 is the angle which the displacement b%
makes with the normal to this parallelogram, the volume of the
parallelepiped is
ds . 33 sin 0 . bx cos 77 — 8 N.
Now X bx = i 5 N = i ds . 33 sin 0 fix cos 77,
and X =. i ds . 33 sin 0 cos 77
is the force which urges ds, resolved in the direction 8#.
The direction of this force is therefore perpendicular to the paral
lelogram, and is equal to i . ds . 33 sin 0.
This is the area of a parallelogram whose sides represent in mag
nitude and direction i ds and 33. The force acting on ds is therefore
represented in magnitude by the area of this parallelogram, and
in direction by a normal to its plane drawn in the direction of the
longitudinal motion of a right-handed screw, the handle of which
138
ELECTROMAGNETIC FORCE.
[491.
South
East
is turned from the direction of the current ids to that of the
magnetic induction 33.
We may express in the language of
Quaternions, both the direction and
West ^ J^ North the magnitude of this force by saying
that it is the vector part of the result
of multiplying the vector ids, the
element of the current, by the vector
33, the magnetic induction.
491.] We have thus completely de
termined the force which acts on any
portion of an electric circuit placed in
a magnetic field. If the circuit is
moved in any way so that, after assuming various forms and
positions, it returns to its original place, the strength of the
current remaining constant during the motion, the whole amount
of work done by the electromagnetic forces will be zero. Since
this is true of any cycle of motions of the circuit, it follows that
it is impossible to maintain by electromagnetic forces a motion
of continuous rotation in any part of a linear circuit of constant
strength against the resistance of friction, &c.
It is possible, however, to produce continuous rotation provided
that at some part of the course of the electric current it passes
from one conductor to another which slides or glides over it.
When in a circuit there is sliding contact of a conductor over
the surface of a smooth solid or
a fluid, the circuit can no longer
be considered as a single linear
circuit of constant strength, but
must be regarded as a system of
two or of some greater number
of circuits of variable strength,
the current being so distributed
among them that those for
which N is increasing have
currents in the positive direc
tion, while those for which N is diminishing have currents in the
negative direction.
Thus, in the apparatus represented in Fig. 23, OP is a moveable
conductor, one end of which rests in a cup of mercury 0, while the
other dips into a circular trough of mercury concentric with 0.
Fig. 23.
492.] CONTINUOUS KOTATION. 139
The current i enters along AB, and divides in the circular trough
into two parts, one of which, #, flows along the arc BQP, while the
other, y, flows along BRP. These currents, uniting at P, flow
along the moveable conductor PO and the electrode OZ to the zinc
end of the battery. The strength of the current along OP and OZ
is x + y or i.
Here we have two circuits, ABQPOZ, the strength of the current
in which is x, flowing in the positive direction, and ABRPOZ, the
strength of the current in which is y> flowing in the negative
direction.
Let 23 be the magnetic induction, and let it be in an upward
direction, normal to the plane of the circle.
While OP moves through an angle 9 in the direction opposite
to that of the hands of a watch, the area of the first circuit increases
by i#P2. 0, and that of the second diminishes by the same quantity.
Since the strength of the current in the first circuit is #, the work
done by it is J x. OP2. 0.33, and since the strength of the second
is — y, the work done by it is \y.OP2. 6 33. The whole work done
is therefore
i(tf + 3/)OP2.033 or ii.OP2.0«B,
depending only on the strength of the current in PO. Hence, if
i is maintained constant, the arm OP will be carried round and
round the circle with a uniform force whose moment is \i .OP2 53.
If, as in northern latitudes, 33 acts downwards, and if the current
is inwards, the rotation will be in the negative direction, that is,
in the direction PQBR.
492.] We are now able to pass from the mutual action of
magnets and currents to the action of one current on another.
For we know that the magnetic properties of an electric circuit C± ,
with respect to any magnetic system M2, are identical with those
of a magnetic shell S19 whose edge coincides with the circuit, and
whose strength is numerically equal to that of the electric current.
Let the magnetic system M2 be a magnetic shell S2, then the
mutual action between ^ and 82 is identical with that between ^
and a circuit C2, coinciding with the edge of S2 and equal in
numerical strength, and this latter action is identical with that
between Ct and C2.
Hence the mutual action between two circuits, Cl and C2) is
identical with that between the corresponding magnetic shells Sl
and S2.
We have already investigated, in Art. 423, the mutual action
140 ELECTROMAGNETIC FORCE. [493-
of two magnetic shells whose edges are the closed curves s1 and s2 .
/**2 /**! COS 6
If we make M= I - &,<&«,
J0 ^0 ?
where e is the angle between the directions of the elements ds1 and
ds2, and r is the distance between them, the integration being
extended once round s.2 and once round slf and if we call M the
potential of the two closed curves ^ and <s2, then the potential energy
due to the mutual action of two magnetic shells whose strengths
are ^ and a'2 bounded by the two circuits is
and the force X, which aids any displacement 8#, is
The whole theory of the force acting on any portion of an electric
circuit due to the action of another electric circuit may be deduced
from this result.
493.] The method which we have followed in this chapter is
that of Faraday. Instead of beginning, as we shall do, following
Ampere, in the next chapter, with the direct action of a portion
of one circuit on a portion of another, we shew, first, that a circuit
produces the same effect on a magnet as a magnetic shell, or, in
other words, we determine the nature of the magnetic field due
to the circuit. We shew, secondly, that a circuit when placed in
any magnetic field experiences the same force as a magnetic shell.
We thus determine the force acting on the circuit placed in any
magnetic field. Lastly, by supposing the magnetic field to be due
to a second electric circuit we determine the action of one circuit
on the whole or any portion of the other.
494.] Let us apply this method to the case of a straight current
of infinite length acting on a portion of a parallel straight con
ductor.
Let us suppose that a current i in the first conductor is flowing
vertically downwards. In this case the end of a magnet which
points north will point to the right-hand of a man looking at it
from the axis of the current.
The lines of magnetic induction are therefore horizontal circles,
having their centres in the axis of the current, and their positive
direction is north, east, south, west.
Let another descending vertical current be placed due west of
the first. The lines of magnetic induction clue to the first current
496.] ELECTROMAGNETIC MEASURE OF A CURRENT. 141
are here directed towards the north. The direction of the force
acting on the second current is to be determined by turning the
handle of a right-handed screw from the nadir, the direction of
the current, to the north, the direction of the magnetic induction.
The screw will then move towards the east, that is, the force acting
on the second current is directed towards the first current, or, in
general, since the phenomenon depends only on the relative position
of the currents, two parallel currents in the same direction attract
each other.
In the same way we may shew that two parallel currents in
opposite directions repel one another.
495.] The intensity of the magnetic induction at a distance r
from a straight current of strength i is, as we have shewn in
Art. 479, i
2-.
r
Hence, a portion of a second conductor parallel to the first, and
carrying a current i' in the same direction, will be attracted towards
the first with a force
where a is the length of the portion considered, and r is its distance
from the first conductor.
Since the ratio of a to r is a numerical quantity independent of
the absolute value of either of these lines, the product of two
currents measured in the electromagnetic system must be of the
dimensions of a force, hence the dimensions of the unit current are
[i] = [F*] = \_M* L* T-*].
496.] Another method of determining the direction of the force
which acts on a current is to consider the relation of the magnetic
action of the current to that of other currents and magnets.
If on one side of the wire which carries the current the magnetic
action due to the current is in the same or nearly the same direction
as that due to other currents, then, on the other side of the wire,
these forces will be in opposite or nearly opposite directions, and
the force acting on the wire will be from the side on which the
forces strengthen each other to the side on which they oppose each
other.
Thus, if a descending current is placed in a field of magnetic
force directed towards the north, its magnetic action will be to the
north on the west side, and to the south on the east side. Hence
the forces strengthen each other on the west side and oppose each
142 ELECTROMAGNETIC FORCE. [497-
other on the east side, and the current will therefore be acted
on by a force from west to east. See Fig. 22, p. 138.
In Fig. XVII at the end of this volume the small circle represents
a section of the wire carrying a descending current, and placed
in a uniform field of magnetic force acting towards the left-hand
of the figure. The magnetic force is greater below the wire than
above it. It will therefore be urged from the bottom towards the
top of the figure.
497.] If two currents are in the same plane but not parallel,
we may apply this principle. Let one of the conductors be an
infinite straight wire in the plane of the paper, supposed horizontal.
On the right side of the current the magnetic force acts downward,
and on the left side it acts upwards. The same is true of the mag
netic force due to any short portion of a second current in the same
plane. If the second current is on the right side of the first, the
magnetic forces will strengthen each other on its right side and
oppose each other on its left side. Hence the second current will
be acted on by a force urging it from its right side to its left side.
The magnitude of this force depends only on the position of the
second current and not on its direction. If the second current is
on the left side of the first it will be urged from left to right.
Hence, if the second current is in the same direction as the first
it is attracted, if in the opposite direction it is repelled, if it flows
at right angles to the first and away from it, it is urged in the
direction of the first current, and if it flows toward the first current,
it is urged in the direction opposite to that in which the first
current flows.
In considering the mutual action of two currents it is not neces
sary to bear in mind the relations between electricity and magnetism
which we have endeavoured to illustrate by means of a right-handed
screw. Even if we have forgotten these relations we shall arrive
at correct results, provided we adhere consistently to one of the two
possible forms of the relation.
498.] Let us now bring together the magnetic phenomena of
the electric circuit so far as we have investigated them.
We may conceive the electric circuit to consist of a voltaic
battery, and a wire connecting its extremities, or of a thermoelectric
arrangement, or of a charged Leyden jar with a wire connecting its
positive and negative coatings, or of any other arrangement for
producing an electric current along a definite path.
The current produces magnetic phenomena in its neighbourhood.
499-] RECAPITULATION. 143
If any closed curve be drawn, and the line-integral of the
magnetic force taken completely round it, then, if the closed curve
is not linked with the circuit, the line-integral is zero, but if it
is linked with the circuit, so that the current i flows through the
closed curve, the line-integral is 4 IT i, and is positive if the direction
of integration round the closed curve would coincide with that
of the hands of a watch as seen by a person passing through it
in the direction in which the electric current flows. To a person
moving along the closed curve in the direction of integration, and
passing through the electric circuit, the direction of the current
would appear to be that of the hands of a watch. We may express
this in another way by saying that the relation between the direc
tions of the two closed curves may be expressed by describing a
right-handed screw round the electric circuit and a right-handed
screw round the closed curve. If the direction of rotation of the
thread of either, as we pass along it, coincides with the positive
direction in the other, then the line-integral will be positive, and
in the opposite case it will be negative.
Fig. 24.
Relation between the electric current and the lines of magnetic induction indicated
by a right-handed screw.
499.] Note. — The line-integral 4 TT i depends solely on the quan
tity of the current, and not on any other thing whatever. It
does not depend on the nature of the conductor through which
the current is passing, as, for instance, whether it be a metal
or an electrolyte, or an imperfect conductor. We have reason
for believing that even when there is no proper conduction, but
144 ELECTROMAGNETIC FORCE. [5OO.
merely a variation of electric displacement, as in the glass of a
Leyden jar during charge or discharge, the magnetic effect of the
electric movement is precisely the same.
Again , the value of the line-integral 4 TT i does not depend on
the nature of the medium in which the closed curve is drawn.
It is the same whether the closed curve is drawn entirely through
air, or passes through a magnet, or soft iron, or any other sub
stance, whether paramagnetic or diamagnetic.
500.] When a circuit is placed in a magnetic field the mutual
action between the current and the other constituents of the field
depends on the surface-integral of the magnetic induction through
any surface bounded by that circuit. If by any given motion of
the circuit, or of part of it, this surface-integral can be increased,
there will be a mechanical force tending to move the conductor
or the portion of the conductor in the given manner.
The kind of motion of the conductor which increases the surface-
integral is motion of the conductor perpendicular to the direction
of the current and across the lines of induction.
If a parallelogram be drawn, whose sides are parallel and pro
portional to the strength of the current at any point, and to the
magnetic induction at the same point, then the force on unit of
length of the conductor is numerically equal to the area of this
parallelogram, and is perpendicular to its plane, and acts in the
direction in which the motion of turning the handle of a right-
handed screw from the direction of the current to the direction
of the magnetic induction would cause the screw to move.
Hence we have a new electromagnetic definition of a line of
magnetic induction. It is that line to which the force on the
conductor is always perpendicular.
It may also be defined as a line along which, if an electric current
be transmitted, the conductor carrying it will experience no force.
501.] It must be carefully remembered, that the mechanical force
which urges a conductor carrying a current across the lines of
magnetic force, acts, not on the electric current, but on the con
ductor which carries it. If the conductor be a rotating disk or a
fluid it will move in obedience to this force, and this motion may
or may not be accompanied with a change of position of the electric
current which it carries. But if the current itself be free to choose
any path through a fixed solid conductor or a network of wires,
then, when a constant magnetic force is made to act on the system,
the path of the current through the conductors is not permanently
501-]
RECAPITULATION.
145
altered, but after certain transient phenomena, called induction
currents, have subsided, the distribution of the current will be found
to be the same as if no magnetic force were in action.
The only force which acts on electric currents is electromotive
force, which must be distinguished from the mechanical force which
is the subject of this chapter.
Fig. 25.
Relations between the positive directions of motion and of rotation indicated by
three right-handed screws.
VOL. II.
CHAPTER II.
AMPERE'S INVESTIGATION OF THE MUTUAL ACTION OF
ELECTRIC CURRENTS.
502.] WE have considered in the last chapter the nature of the
magnetic field produced by an electric current; and the mechanical
action on a conductor carrying an electric current placed in a mag
netic field. From this we went on to consider the action of one
electric circuit upon another, by determining the action on the first
due to the magnetic field produced by the second. But the action
of one circuit upon another was originally investigated in a direct
manner by Ampere almost immediately after the publication of
Orsted's discovery. We shall therefore give an outline of Ampere's
method, resuming the method of this treatise in the next chapter.
The ideas which guided Ampere belong to the system which
admits direct action at a distance, and we shall find that a remark
able course of speculation and investigation founded on these ideas
has been carried on by Gauss, Weber, J. Neumann, Riemann,
Betti, C. Neumann, Lorenz, and others, with very remarkable
results both in the discovery of new facts and in the formation of
a theory of electricity. See Arts. 846-866.
The ideas which I have attempted to follow out are those of
action through a medium from one portion to the contiguous
portion. These ideas were much employed by Faraday, and the
development of them in a mathematical form, and the comparison of
the results with known facts, have been my aim in several published
papers. The comparison, from a philosophical point of view, of the
results of two methods so completely opposed in their first prin
ciples must lead to valuable data for the study of the conditions
of scientific speculation.
503.] Ampere's theory of the mutual action of electric currents
is founded on four experimental facts and one assumption.
505.] AMPERE'S SCIENTIFIC METHOD. 147
Ampere's fundamental experiments are all of them examples of
what has been called the null method of comparing' forces. See
Art. 214. Instead of measuring the force by the dynamical effect
of communicating1 motion to a body, or the statical method of
placing it in equilibrium with the weight of a body or the elasticity
of a fibre, in the null method two forces, due to the same source,
are made to act simultaneously on a body already in equilibrium,
and no effect is produced, which shews that these forces are them
selves in equilibrium. This method is peculiarly valuable for
comparing the effects of the electric current when it passes through
circuits of different forms. By connecting all the conductors in
one continuous series, we ensure that the strength of the current
is the same at every point of its course, and since the current
begins everywhere throughout its course almost at the same instant,
we may prove that the forces due to its action on a suspended
body are in equilibrium by observing that the body is not at all
affected by the starting or the stopping of the current.
504.] Ampere's balance consists of a light frame capable of
revolving1 about a vertical axis, and carrying1 a wire which forms
two circuits of equal area, in the same plane or in parallel planes,
in which the current flows in opposite directions. The object of
this arrangement is to get rid of the effects of terrestrial magnetism
on the conducting wire. When an electric circuit is free to move
it tends to place itself so as to embrace the largest possible number
of the lines of induction. If these lines are due to terrestrial
magnetism, this position, for a circuit in a vertical plane, will be
when the plane of the circuit is east and west, and when the
direction of the current is opposed to the apparent course of the
sun.
By rigidly connecting two circuits of equal area in parallel planes,
in which equal currents run in opposite directions, a combination
is formed which is unaffected by terrestrial magnetism, and is
therefore called an Astatic Combination, see Fig. 26. It is acted
on, however, by forces arising from currents or magnets which are
so near it that they act differently on the two circuits.
505.] Ampere's first experiment is on the effect of two equal
currents close together in opposite directions. A wire covered with
insulating material is doubled on itself, and placed near one of the
circuits of the astatic balance. When a current is made to pass
through the wire and the balance, the equilibrium of the balance
remains undisturbed, shewing that two equal currents close together
L 2
148
AMPERES THEORY.
[506.
in opposite directions neutralize each other. If, instead of two
wires side by side, a wire be insulated in the middle of a metal
Fig. 26.
tube, and if the current pass through the wire and back by the
tube, the action outside the tube is not only approximately but
accurately null. This principle is of great importance in the con
struction of electric apparatus, as it affords the means of conveying
the current to and from any galvanometer or other instrument in
such a way that no electromagnetic effect is produced by the current
on its passage to and from the instrument. In practice it is gene
rally sufficient to bind the wires together, care being taken that
they are kept perfectly insulated from each other, but where they
must pass near any sensitive part of the apparatus it is better to
make one of the conductors a tube and the other a wire inside it.
See Art. 683.
506.] In Ampere's second experiment one of the wires is bent
and crooked with a number of small sinuosities, but so that in
every part of its course it remains very near the straight wire.
A current, flowing through the crooked wire and back again
through the straight wire, is found to be without influence on the
astatic balance. This proves that the effect of the current running
through any crooked part of the wire is equivalent to the same
current running in the straight line joining its extremities, pro
vided the crooked line is in no part of its course far from the
straight one. Hence any small element of a circuit is equivalent
to two or more component elements, the relation between the
component elements and the resultant element being the same as
that between component and resultant displacements or velocities.
507.] In the third experiment a conductor capable of moving
508.]
FOUK EXPERIMENTS.
149
only in the direction of its length is substituted for the astatic
balance, the current enters the conductor and leaves it at fixed
points of space, and it is found that no closed circuit placed in
the neighbourhood is able to move the conductor.
Fig. 27.
The conductor in this experiment is a wire in the form of a
circular arc suspended on a frame which is capable of rotation
about a vertical axis. The circular arc is horizontal, and its centre
coincides with the vertical axis. Two small troughs are filled with
mercury till the convex surface of the mercury rises above the
level of the troughs. The troughs are placed under the circular
arc and adjusted till the mercury touches the wire, which is of
copper well amalgamated. The current is made to enter one of
these troughs, to traverse the part of the circular arc between the
troughs, and to escape by the other trough. Thus part of the
circular arc is traversed by the current, and the arc is at the same
time capable of moving with considerable freedom in the direc
tion of its length. Any closed currents or magnets may now be
made to approach the moveable conductor without producing the
slightest tendency to move it in the direction of its length.
508.] In the fourth experiment with the astatic balance two
circuits are employed, each similar to one of those in the balance,
but one of them, C, having dimensions n times greater, and the
other, A, n times less. These are placed on opposite sides of the
circuit of the balance, which we shall call B, so that they are
similarly placed with respect to it, the distance of C from B being
n times greater than the distance of B from A. The direction and
150
AMPERES THEORY.
[5o8.
strength of the current is the same in A and C. Its direction in
B may be the same or opposite. Under these circumstances it is
found that B is in equilibrium under the action of A and C, whatever
be the forms and distances of the three circuits, provided they have
the relations given above.
Since the actions between the complete circuits may be considered
to be due to actions between the elements of the circuits, we may
use the following method of determining the law of these actions.
Let Alt BI} Cv Fig. 28, be corresponding elements of the three
circuits, and let A2, B2, C2 be also corresponding elements in an
other part of the circuits. Then the situation of B± with respect
to A2 is similar to the situation of C^ with respect to B.2) but the
0
u
distance and dimensions of Cl and B2 are n times the distance and
dimensions of Bl and A2i respectively. If the law of electromag
netic action is a function of the distance, then the action, what
ever be its form or quality, between Bl and A.2, may be written
and that between C1 and B2
'
where #, b, c are the strengths of the currents in A, B, C. But
A — CB and a = c. Hence
^ = Clt
= B
and this is equal to F by experiment, so that we have
or, the force varies inversely as the square of the distance.
511.] FOKCE- BETWEEN TWO ELEMENTS. 151
509.] It may be observed with reference to these experiments
that every electric current forms a closed circuit. The currents
used by Ampere, being produced by the voltaic battery, were of
course in closed circuits. It might be supposed that in the case
of the current of discharge of a conductor by a spark we might
have a current forming an open finite line, but according to the
views of this book even this case is that of a closed circuit. No
experiments on the mutual action of unclosed currents have been
made. Hence no statement about the mutual action of two ele
ments of circuits can be said to rest on purely experimental grounds.
It is true we may render a portion of a circuit moveable, so as to
ascertain the action of the other currents upon it, but these cur
rents, together with that in the moveable portion, necessarily form
closed circuits, so that the ultimate result of the experiment is the
action of one or more closed currents upon the whole or a part of a
closed current.
510.] In the analysis of the phenomena, however, we may re
gard the action of a closed circuit on an element of itself or of
another circuit as the resultant of a number of separate forces,
depending on the separate parts into which the first circuit may
be conceived, for mathematical purposes, to be divided.
This is a merely mathematical analysis of the action, and is
therefore perfectly legitimate, whether these forces can really act
separately or not.
511.] We shall begin by considering the purely geometrical
relations between two lines in space representing the circuits, and
between elementary portions of these lines.
Let there be two curves in space in each of which a fixed point
is taken from which the arcs are
measured in a defined direction
along the curve. Let A, A' be
these points. Let PQ and P Q'
be elements of the two curves.
Let AP=s, A'P'=s
and let the distance PPf be de- Fig< 29'
noted by r. Let the angle P*PQ be denoted by 0, and PP'(g
by Qf, and let the angle between the planes of these angles be
denoted by rj.
The relative position of the two elements is sufficiently defined by
their distance r and the three angles 0, 6' , and r/, for if these be
152
AMPERES THEORY.
given their relative position is as completely determined as if they
formed part of the same rigid body.
512.] If we use rectangular coordinates and make #, y, z the
coordinates of P, and of, y', z' those of P', and if we denote by I, m,
n and by I', m', n' the direction-cosines of PQ, and of P'Q' re
spectively, then
dx j dy dz -»
-J-—1) -f- = m, = n,
as as as
dx' ,, dy' , dz' ,
(2)
and I \x' — x) + m (y' — y} + n (z' — z) = rcos0,
I' (xf — x] -f- m' (y' — y) -f n (zf — z) = — rcos6\ (3)
II' -f mm' -f nn' = cos e,
where e is the angle between the directions of the elements them
selves, and
cos e = — cos 6 cos 6' + sin 0 sin (f cos rj. (4)
r* = (af—x)* + (tf—y)* + (af-z)2, (5)
Again
.
whence
+ -*,
dr . , . dx , , N dy , , . dz
- = -(* -*) _(y _,) -(, -z)
= — rcosO.
dr
Similarly r= (^-
. i . <
-^) +(/-*)
\ (6)
= — r cos 6 ;
and differentiating r -=- with respect to /,
dr dr dx dx' dy dy dz dz'
CvS CtS CvS CvS CvS CvS CtS dS
(7)
— — (II' -j- mm' + n n'}
= — cos e. j
We can therefore express the three angles 0, 6', and r;, and the
auxiliary angle e in terms of the differential coefficients of r with
respect to s and s' as follows,
dr
cos 0 =
dr
cose = — r
dr dr
d2r
sin 6 sin 6' cos 77 = — r -
(8)
513-] GEOMETRICAL RELATIONS OF TWO ELEMENTS. 153
513.] We shall next consider in what way it is mathematically
conceivable that the elements PQ and PQ' might act on each
other, and in doing so we shall not at first assume that their mutual
action is necessarily in the line joining them.
We have seen that we may suppose each element resolved into
other elements, provided that these components, when combined
according to the rule of addition of vectors, produce the original
element as their resultant.
We shall therefore consider ds as resolved into cos 6 ds — a in the
direction of r, and sin 6 ds = /3 fl ^
in a direction perpendicular to \ / *^\/
T in the plane P'PQ. p « «'>"'
We shall also consider ds'
as resolved into cos Q' els' = a in the direction of r reversed,
mntfoO8ri(tf=P in a direction parallel to that in which /3 was
measured, and sin 0' sin 17 els' = y in a direction perpendicular to
a and /3'.
Let us consider the action between the components a and j3 on
the one hand, and a, /3', / on the other.
(1) a and a are in the same straight line. The force between
them must therefore be in this line. We shall suppose it to be
an attraction = Aa<xii't
where A is a function of r, and i} i' are the intensities of the
currents in ds and els' respectively. This expression satisfies the
condition of changing sign with i and with i'm
(2) /3 and (3' are parallel to each other and perpendicular to the
line joining them. The action between them may be written
This force is evidently in the line joining (3 and /3', for it must
be in the plane in which they both lie, and if we were to measure
(3 and ft in the reversed direction, the value of this expression
would remain the same, which shews that, if it represents a force,
that force has no component in the direction of f3, and must there
fore be directed along r. Let us assume that this expression, when
positive, represents an attraction.
(3) /3 and y are perpendicular to each other and to the line
joining them. The only action possible between elements so related
is a couple whose axis is parallel to T. We are at present engaged
with forces, so we shall leave this out of account.
(4) The action of a and /3', if they act on each other, must be
expressed by
154 AMPERE'S THEORY.
The sign of this expression is reversed if we reverse the direction
in which we measure j3'. It must therefore represent either a force
in the direction of ft', or a couple in the plane of a and /3'. As we
are not investigating couples, we shall take it as a force acting
on a in the direction of ft'.
There is of course an equal force acting on /3' in the opposite
direction.
We have for the same reason a force
Cay'ii'
acting on a in the direction of y', and a force
acting on /3 in the opposite direction.
514.] Collecting our results, we find that the action on ds is
compounded of the following forces,
X = (Aaa' + B (3fi')ii' in the direction of r,
Y— C(a(B' — aj3)ii' in the direction of (3, (9)
and Z — C ay ii' in the direction of y'.
Let us suppose that this action on ds is the resultant of three
forces, Rii'dsds' acting in the direction of r, Sii'dsds' acting in
the direction of ds, and S'ii'dsds' acting in the direction of ds' ,
then in terms of 6, d', and 77,
R = A cos 0 cos 0' + J9sin0sin0'cosr7,
In terms of the differential coefficients of
.r o, r
^ + G-yyJ & = — G--1
ds ds J
In terms of I, m, n, and I', m', n'9
R =-
where f, ??, fare written for af—x, y' — y, and / — z respectively.
515.] We have next to calculate the force with which the finite
current / acts on the finite current s. The current s extends from
A, where s = 0, to P, where it has the value s. The current /
extends from A', where s'= 0, to P', where it has the value /.
5 1 6.] ACTION OF A CLOSED CIRCUIT ON AN ELEMENT. 155
The coordinates of points on either current are functions of s or
of /.
If F is any function of the position of a point, then We shall use
the subscript (s o) to denote the excess of its value at P over that
at A, thus jr(SiQ} = FP-FA,
Such functions necessarily disappear when the circuit is closed.
Let the components of the total force with which A' P* acts on
A A be iif Xj ii'Y, and ii'Z. Then the component parallel to X of
the force with which da' acts on ds will be ii' - — 7-7 da ds'.
dsds
Hence -T = R+8l+8'l'. (13)
r
Substituting the values of R, S, and S' from (12), remembering
(14)
and arranging the terms with respect to lt m, n, we find
ds
Since A, B, and C are functions of r, we may write
P = f (A + £)~dr, Q=[ Cdr, (16)
jr r jr
the integration being taken between r and oo because A, JB, C
vanish when r = oo.
Hence (A + £)-L = -~, and <? = -^. (17)
516.] Now we know, by Ampere's third case of equilibrium, that
when / is a closed circuit, the force acting on ds is perpendicular
to the direction of ds, or, in other words, the component of the force
in the direction of ds itself is zero. Let us therefore assume the
direction of the axis of x so as to be parallel to ds by making I = 1 ,
m — 0, n == 0. Equation (15) then becomes
-
To find — , the force on ds referred to unit of length, we must
ds
156 AMPERE'S THEORY. [5J7-
integrate this expression with respect to /. Integrating the first
term by parts, we find
*X=(Pp-Q)Va-£(2Pr-3-O?l-<U'. (19)
When / is a closed circuit this expression must be zero. The
first term will disappear of itself. The second term, however, will
not in general disappear in the case of a closed circuit unless the
quantity under the sign of integration is always zero. Hence, to
satisfy Ampere's condition,
(20)
517.] We can now eliminate P, and find the general value of
When / is a closed circuit the first term of this expression
vanishes, and if we make
(22)
& r
/=r
JQ Z T
where the integration is extended round the closed circuit /, we
may write c^
Similarly =na'_iyt (23)
u/s
dZ
-j-=lp
ds
The quantities a', (3', y are sometimes called the determinants of
the circuit / referred to the point P. Their resultant is called by
Ampere the directrix of the electrodynamic action.
It is evident from the equation, that the force whose components
dX dY . dZ .
are -^—> -=-, and ~ is perpendicular both to ds and to this
as as ds
directrix, and is represented numerically by the area of the parallel
ogram whose sides are ds and the directrix.
5 1 9-] FORCE BETWEEN TWO FINITE CURRENTS. 157
In the language of quaternions, the resultant force on ds is the
vector part of the product of the directrix multiplied by ds.
Since we already know that the directrix is the same thing as
the magnetic force due to a unit current in the circuit /, we shall
henceforth speak of the directrix as the magnetic force due to the
circuit.
518.] We shall now complete the calculation of the components
of the force acting between two finite currents, whether closed or
open.
Let p be a new function of r, such that
"oo
P = i/ (B-C)dr, (24)
then by (17) and (20)
d% d
and equations (11) become
& — - ' ' au&(Q+P)
B ** , ** \
O = — ^7-7 > O = — - •
ds ds J
With these values of the component forces, equation (13) becomes
l_L_ I' _UL . (27}
w ds ds' ds' ds
519.] Let
F = I Ipds, G = I mpds, H = I npds, (28)
i/O JQ JQ
F = f'l'p ds', G' = f'm'pds', H'= [''n'pds'. (29)
^0 "'0 Jo
These quantities have definite values for any given point of space.
When the circuits are closed, they correspond to the components of
the vector-potentials of the circuits.
Let L be a new function of r, such that
fr
L — I r(Q + p)dr, (30)
^o
and let M be the double integral
M = I I pcosedsds', (31)
«^0 * 0
158 AMPERE'S THEORY. [520.
which, when the circuits are closed, becomes their mutual potential,
then (27) may be written
«•„•* \dM dL }
dsds'~ dsds I da dx^ \
520.] Integrating1, with respect to s and *', between the given
limits, we find
d_
dx dx
+ F'P-FfA-FP, + FAf, (33)
where the subscripts of L indicate the distance, r, of which the
quantity L is a function, and the subscripts of F and F' indicate
the points at which their values are to be taken.
The expressions for Y and Z may be written down from this.
Multiplying the three components by dxt dy, and dz respectively,
we obtain
Xdx+Ydy + Zdz = DM-D(Lpp,—LAP,—LA,p
X — — — =- (LPp> — LAP'— LA' p
P,-Ay, (34)
where D is the symbol of a complete differential.
Since Fdx + Gdy + Hdz is not in general a complete differential of
a function of #,y, £, Xdx + Ydy + Zdz is not a complete differential
for currents either of which is not closed.
521.] If, however, both currents are closed, the terms in I/, F,
G, H, F, G't H' disappear, and
Xdx+Ydy + Zdz = DM, (35)
where M is the mutual potential of two closed circuits carrying unit
currents. The quantity M expresses the work done by the electro
magnetic forces on either conducting circuit when it is moved
parallel to itself from an infinite distance to its actual position. Any
alteration of its position, by which M is increased, will be assisted by
the electromagnetic forces.
It may be shewn, as in Arts. 490, 596, that when the motion of
the circuit is not parallel to itself the forces acting on it are still
determined by the variation of M, the potential of the one circuit on
the other.
522.] The only experimental fact which we have made use of
in this investigation is the fact established by Ampere that the
action of a closed current on any portion of another current is
perpendicular to the direction of the latter. Every other part of
524.] HIS FORMULA. 159
the investigation depends on purely mathematical considerations
depending on the properties of lines in space. The reasoning there
fore may be presented in a much more condensed and appropriate
form by the use of the ideas and language of the mathematical
method specially adapted to the expression of such geometrical
relations — the Quaternions of Hamilton.
This has been done by Professor Tait in the Quarterly Mathe
matical Journal, 1866, and in his treatise on Quaternions, § 399, for
Ampere's original investigation, and the student can easily adapt
the same method to the somewhat more general investigation given
here.
523.] Hitherto we have made no assumption with respect to the
quantities A, B, C, except that they are functions of r, the distance
between the elements. We have next to ascertain the form of
these functions, and for this purpose we make use of Ampere's
fourth case of equilibrium, Art. 508, in which it is shewn that if
all the linear dimensions and distances of a system of two circuits
be altered in the same proportion, the currents remaining the same,
the force between the two circuits will remain the same.
Now the force between the circuits for unit currents is -= — , and
dos
since this is independent of the dimensions of the system, it must
be a numerical quantity. Hence M itself, the coefficient of the
mutual potential of the circuits, must be a quantity of the dimen
sions of a line. It follows, from equation (31), that p must be the
reciprocal of a line, and therefore by (24), B— (7 must be the inverse
square of a line. But since B and C are both functions of r, B—C
must be the inverse square of r or some numerical multiple of it.
524.] The multiple we adopt depends on our system of measure
ment. If we adopt the electromagnetic system, so called because
it agrees with the system already established for magnetic measure
ments, the value of M ought to coincide with that of the potential
of two magnetic shells of strength unity whose boundaries are the
two circuits respectively. The value of M in that case is, by
Art. 423, /"/"cos* ,
M = J I - ds ds', (36)
the integration being performed round both circuits in the positive
direction. Adopting this as the numerical value of M, and com
paring with (31), we find
p = , and S-C=~. (37)
160 AMPERE'S THEORY. [525-
525.] We may now express the components of the force on ds
arising from the action of ds' in the most general form consistent
with experimental facts.
The force on ds is compounded of an attraction
1 /dr dr d2r \ . d2 0 . 7 , 1
R = -H- l-y- -=-, — 2r -7-77) ^^ dsds' -f r -^—j-,11 ds ds
rz ^ds ds dsds' dsds
in the direction of r,
S = 77 i i'ds ds in the direction of ds,
as
and S' = —^ ii'ds ds' in the direction of ds'.
ds
/NO
where Q — / Cdr, and since C is an unknown function of r, we
J r
know only that Q is some function of r.
526.] The quantity Q cannot be determined, without assump
tions of some kind, from experiments in which the active current
forms a closed circuit. If we suppose with Ampere that the action
between the elements ds and ds' is in the line joining them, then
S and 8' must disappear, and Q must be constant, or zero. The
force is then reduced to an attraction whose value is
(39)
Ampere, who made this investigation long before the magnetic
system of units had been established, uses a formula having a
numerical value half of this, namely
1 A dr dr dr N . ., _
R = -2 (- -7- -T7 - r -j—r^Jjds ds'. (40)
f2 \9 fix Of d*njfJ** v '
Here the strength of the current is measured in what is called
electro dynamic measure. If i, i' are the strength of the currents in
electromagnetic measure, and j, j' the same in electrodynamic mea
sure, then it is plain that
jf = 2ii', or j = ^i. (41)
Hence the unit current adopted in electromagnetic measure is
greater than that adopted in electrodynamic measure in the ratio
of «/2 to 1.
The only title of the electrodynamic unit to consideration is
that it was originally adopted by Ampere, the discoverer of the
law of action between currents. The continual recurrence of <s/2
in calculations founded on it is inconvenient, and the electro
magnetic system has the great advantage of coinciding numerically
527.] FOUK ASSUMPTIONS. 161
with all our magnetic formulae. As it is difficult for the student
to bear in mind whether he is to multiply or to divide by \/2, we
shall henceforth use only the electromagnetic system, as adopted by
Weber and most other writers.
Since the form and value of Q have no effect on any of the
experiments hitherto made, in which the active current at least
is always a closed one, we may, if we please, adopt any value of Q
which appears to us to simplify the formulae.
Thus Ampere assumes that the force between two elements is in
the line joining them. This gives Q = 0,
(42)
r
Grassmann * assumes that two elements in the same straight line
have no mutual action. This gives
Q 1 R- 3 d*T 8- l dr 8'- l - (43)
V= ~2~r' ~Trdsds" 2r* els'3 ~ 2r* ds ( }
We might, if we pleased, assume that the attraction between two
elements at a given distance is proportional to the cosine of the
angle between them. In this case
„ 1 _ 1 0 1 dr 0, 1 dr , . .
«=--> JZ = ^«»c, * = -F5p. S'=^Ts. (44)
Finally, we might assume that the attraction and the oblique
forces depend only on the angles which the elements make with the
line joining them, and then we should have
0- 2 R- 3ldrdr S- 2- S'-*~. (45)
V* ~P * VS3?1 ~PdS> ~ r* ds ( }
527.] Of these four different assumptions that of Ampere is
undoubtedly the best, since it is the only one which makes the
forces on the two elements not only equal and opposite but in the
straight line which joins them.
* Pogg., Ann. Ixiv. p. 1 (1845).
VOL. II. M
CHAPTER III
ON THE INDUCTION OF ELECTRIC CURRENTS.
528.] THE discovery by Orsted of the magnetic action of an
electric current led by a direct process of reasoning to that of
magnetization by electric currents, and of the mechanical action
between electric currents. It was not, however, till 1831 that
Faraday, who bad been for some time endeavouring to produce
electric currents by magnetic or electric action, discovered the con
ditions of magneto-electric induction. The method which Faraday
employed in his researches consisted in a constant appeal to ex
periment as a means of testing the truth of his ideas, and a constant
cultivation of ideas under the direct influence of experiment. In
his published researches we find these ideas expressed in language
which is all the better fitted for a nascent science, because it is
somewhat alien from the style of physicists who have been accus
tomed to established mathematical forms of thought.
The experimental investigation by which Ampere established the
laws of the mechanical action between electric currents is one of
the most brilliant achievements in science.
The whole, theory and experiment, seems as if it had leaped,
full grown and full armed, from the brain of the ' Newton of elec
tricity.' It is perfect in form, and unassailable in accuracy, and
it is summed up in a formula from which all the phenomena may
be deduced, and which must always remain the cardinal formula of
electro-dynamics.
The method of Ampere, however, though cast into an inductive
form, does not allow us to trace the formation of the ideas which
guided it. We can scarcely believe that Ampere really discovered
the law of action by means of the experiments which he describes.
We are led to suspect, what, indeed, he tells us himself*, that he
* Theorie des Phenomenes Elect rodynamiqucs, p. 9.
529.] ' FARADAY'S SCIENTIFIC METHOD. 163
discovered the law by some process which he has not shewn us,
and that when he had afterwards built up a perfect demon
stration he removed all traces of the scaffolding by which he had
raised it.
Faraday, on the other hand, shews us his unsuccessful as well
as his successful experiments, and his crude ideas as well as his
developed ones, and the reader, however inferior to him in inductive
power, feels sympathy even more than admiration, and is tempted
to believe that, if he had the opportunity, he too would be a dis
coverer. Every student therefore should read Ampere's research
as a splendid example of scientific style in the statement of a dis
covery, but he should also study Faraday for the cultivation of a
scientific spirit, by means of the action and reaction which will
take place between newly discovered facts and nascent ideas in his
own mind.
It was perhaps for the advantage of science that Faraday, though
thoroughly conscious of the fundamental forms of space, time, and
force, was not a professed mathematician. He was not tempted
to enter into the many interesting researches in pure mathematics
which his discoveries would have suggested if they had been
exhibited in a mathematical form, and he did not feel called upon
either to force his results into a shape acceptable to the mathe
matical taste of the time, or to express them in a form which
mathematicians might attack. He was thus left at leisure to
do his proper work, to coordinate his ideas with his facts, and to
express them in natural, untechnical language.
It is mainly with the hope of making these ideas the basis of a
mathematical method that I have undertaken this treatise.
529.] We are accustomed to consider the universe as made up of
parts, and mathematicians usually begin by considering a single par
ticle, and then conceiving its relation to another particle, and so on.
This has generally been supposed the most natural method. To
conceive of a particle, however, requires a process of abstraction,
since all our perceptions are related to extended bodies, so that
the idea of the all that is in our consciousness at a given instant
is perhaps as primitive an idea as that of any individual thing.
Hence there may be a mathematical method in which we proceed
from the whole to the parts instead of from the parts to the whole.
For example, Euclid, in his first book, conceives a line as traced
out by a point, a surface as swept out by a line, and a solid as
generated by a surface. But he also defines a surface as the
M 2
164 MAGNETO-ELECTRIC INDUCTION,* [530.
boundary of a solid, a line as the edge of a surface, and a point
as the extremity of a line.
In like manner we may conceive the potential of a material
system as a function found by a certain process of integration with
respect to the masses of the bodies in the field, or we may suppose
these masses themselves to have no other mathematical meaning
than the volume-integrals of — V2^? where ^ is the potential.
In electrical investigations we may use formulae in which the
quantities involved are the distances of certain bodies, and the
electrifications or currents in these bodies, or we may use formulae
which involve other quantities, each of which is continuous through
all space.
The mathematical process employed in the first method is in
tegration along lines, over surfaces, and throughout finite spaces,
those employed in the second method are partial differential equa
tions and integrations throughout all space.
The method of Faraday seems to be intimately related to the
second of these modes of treatment. He never considers bodies
as existing with nothing between them but their distance, and
acting on one another according to some function of that distance.
He conceives all space as a field of force, the lines of force being
in general curved, and those due to any body extending from it on
all sides, their directions being modified by the presence of other
bodies. He even speaks * of the lines of force belonging to a body
as in some sense part of itself, so that in its action on distant
bodies it cannot be said to act where it is not. This, however,
is not a dominant idea with Faraday. I think he would rather
have said that the field of space is full of lines of force, whose
arrangement depends on that of the bodies in the field, and that
the mechanical and electrical action on each body is determined by
the lines which abut on it.
PHENOMENA OF MAGNETO-ELECTRIC INDUCTION f.
530.] 1. Induction by Variation of the Primary Current.
Let there be two conducting circuits, the Primary and the
Secondary circuit. The primary circuit is connected with a voltaic
* Exp. Res., ii. p. 293 ; iii. p. 447.
t Read Faraday's Experimental Researches, series i and ii.
530.] ELEMENTARY PHENOMENA. 165
battery by which the primary current may be produced, maintained,
stopped, or reversed. The secondary circuit includes a galvano
meter to indicate any currents which may be formed in it. This
galvanometer is placed at such a distance from all parts of the
primary circuit that the primary current has no sensible direct
influence on its indications.
Let part of the primary circuit consist of a straight wire, and
part of the secondary circuit of a straight wire near, and parallel to
the first, the other parts of the circuits being at a greater distance
from each other.
It is found that at the instant of sending a current through
the straight wire of the primary circuit the galvanometer of the
secondary circuit indicates a current in the secondary straight wire
in the opposite direction. This is called the induced current. If
the primary current is maintained constant, the induced current soon
disappears, and the primary current appears to produce no effect
on the secondary circuit. If now the primary current is stopped,
a secondary current is observed, which is in the same direction as
the primary current. Every variation of the primary current
produces electromotive force in the secondary circuit. When the
primary current increases, the electromotive force is in the opposite
direction to the current. When it diminishes, the electromotive
force is in the same direction as the current. When the primary
current is constant, there is no electromotive force.
These effects of induction are increased by bringing the two wires
nearer together. They are also increased by forming them into
two circular or spiral coils placed close together, and still more by
placing an iron rod or a bundle of iron wires inside the coils.
2. Induction ~by Motion of the Primary Circuit.
We have seen that when the primary current is maintained
constant and at rest the secondary current rapidly disappears.
Now let the primary current be maintained constant, but let the
primary straight wire be made to approach the secondary straight
wire. During the approach there will be a secondary current in
the opposite direction from the primary.
If the primary circuit be moved away from the secondary, there
will be a secondary current in the same direction as the primary.
3. Induction by Motion of the Secondary Circuit.
If the secondary circuit be moved, the secondary current is
166 MAGNETO-ELECTRIC INDUCTION. [S31-
opposite to the primary when the secondary wire is approaching-
the primary wire, and in the same direction when it is receding-
from it.
In all cases the direction of the secondary current is such that
the mechanical action between the two conductors is opposite to
the direction of motion, being a repulsion when the wires are ap
proaching, and an attraction when they are receding. This very
important fact was established by Lenz *.
4. Induction by the Relative Motion of a Magnet and the Secondary
Circuit.
If we substitute for the primary circuit a magnetic shell, whose
edge coincides with the circuit, whose strength is numerically equal
to that of the current in the circuit, and whose austral face cor
responds to the positive face of the circuit, then the phenomena
produced by the relative motion of this shell and the secondary
circuit are the same as those observed in the case of the primary
circuit.
531.] The whole of these phenomena may be summed up in one
law. When the number of lines of magnetic induction which pass
through the secondary circuit in the positive direction is altered,
an electromotive force acts round the circuit, which is measured
by the rate of decrease of the magnetic induction through the
circuit.
532.] For instance, let the rails of a railway be insulated from
the earth, but connected at one terminus through a galvanometer,
and let the circuit be completed by the wheels and axle of a rail
way carriage at a distance x from the terminus. Neglecting the
height of the axle above the level of the rails, the induction
through the secondary circuit is due to the vertical component of
the earth's magnetic force, which in northern latitudes is directed
downwards. Hence, if b is the gauge of the railway, the horizontal
area of the circuit is bx, and the surface-integral of the magnetic
induction through it is Zbxt where Z is the vertical component of
the magnetic force of the earth. Since Z is downwards, the lower
face of the circuit is to be reckoned positive, and the positive
direction of the circuit itself is north, east, south, west, that is, in
the direction of the sun's apparent diurnal course.
Now let the carriage be set in motion, then x will vary, and
* Pogg., Ann. xxi. 483 (1834.)
533-] DIRECTION OF THE FORCE. 167
there will be an electromotive force in the circuit whose value
„, das
is — Zb -=-.
dt
If x is increasing, that is, if the carriage is moving away from
the terminus, this electromotive force is in the negative direction,
or north, west, south, east. Hence the direction of this force
through the axle is from right to left. If x were diminishing, the
absolute direction of the force would be reversed, but since the
direction of the motion of the carriage is also reversed, the electro
motive force on the axle is still from right to left, the observer
in the carriage being always supposed to move face forwards. In
southern latitudes, where the south end of the needle dips, the
electromotive force on a moving body is from left to right.
Hence we have the following rule for determining the electro
motive force on a wire moving through a field of magnetic force.
Place, in imagination, your head and feet in the position occupied
by the ends of a compass needle which point north and south respec
tively ; turn your face in the forward direction of motion, then the
electromotive force due to the motion will be from left to right.
533.] As these directional relations are important, let us take
another illustration. Suppose a metal girdle laid round the earth
at the equator, and a metal wire
laid along the meridian of Green
wich from the equator to the north
pole. /
Let a great quadrantal arch of r/A
metal be constructed, of which one
extremity is pivoted on the north
pole, while the other is carried round
the equator, sliding on the great
girdle of the earth, and following
the sun in his daily course. There
will then be an electromotive force
along the moving quadrant, acting
from the pole towards the equator.
The electromotive force will be the same whether we suppose
the earth at rest and the quadrant moved from east to west, or
whether we suppose the quadrant at rest and the earth turned from
west to east. If we suppose the earth to rotate, the electromotive
force will be the same whatever be the form of the part of the
circuit fixed in space of which one end touches one of the pole&
168 MAGNETO-ELECTRIC INDUCTION. [534-
and the other the equator. The current in this part of the circuit
is from the pole to the equator.
The other part of the circuit, which is fixed with respect to the
earth, may also be of any form, and either within or without the
earth. In this part the current is from the equator to either pole.
534.] The intensity of the electromotive force of magneto -electric
induction is entirely independent of the nature of the substance
of the conductor in which it acts, and also of the nature of the
conductor which carries the inducing current.
To shew this, Faraday * made a conductor of two wires of different
metals insulated from one another by a silk covering, but twisted
together, and soldered together at one end. The other ends of the
wires were connected with a galvanometer. In this way the wires
were similarly situated with respect to the primary circuit, but if
the electromotive force were stronger in the one wire than in the
other it would produce a current which would be indicated by the
galvanometer. He found, however, that such a combination may
be exposed to the most powerful electromotive forces due to in
duction without the galvanometer being affected. He also found
that whether the two branches of the compound conductor consisted
of two metals, or of a metal and an electrolyte, the galvanometer
was not affected f.
Hence the electromotive force on any conductor depends only on
the form and the motion of that conductor, together with the
strength, form, and motion of the electric currents in the field.
535.] Another negative property of electromotive force is that
it has of itself no tendency to cause the mechanical motion of any
body, but only to cause a current of electricity within it.
If it actually produces a current in the body, there will be
mechanical action due to that current, but if we prevent the
current from being formed, there will be no mechanical action on
the body itself. If the body is electrified, however, the electro
motive force will move the body, as we have described in Electro
statics.
536.] The experimental investigation of the laws of the induction
of electric currents in fixed circuits may be conducted with
considerable accuracy by methods in which the electromotive force,
and therefore the current, in the galvanometer circuit is rendered
zero.
For instance, if we wish to shew that the induction of the coil
* Rrp. fas., 195. f Ib., 200.
536.]
EXPERIMENTS OF COMPARISON.
169
A on the coil X is equal to that of B upon Y, we place the first
pair of coils A and X at a sufficient distance from the second pair
Fig. 32.
£ and Y. We then connect A and B with a voltaic battery, so
that we can make the same primary current flow through A in the
positive direction and then through B in the negative direction.
We also connect X and Y with a galvanometer, so that the secondary
current, if it exists, shall flow in the same direction through X and
Yin series.
Then, if the induction of A on X is equal to that of B on Y,
the galvanometer will indicate no induction current when the
battery circuit is closed or broken.
The accuracy of this method increases with the strength of the
primary current and the sensitiveness of the galvanometer to in
stantaneous currents, and the experiments are much more easily
performed than those relating to electromagnetic attractions, where
the conductor itself has to he delicately suspended.
A very instructive series of well devised experiments of this kind
is described by Professor Felici of Pisa *.
I shall only indicate briefly some of the laws which may be proved
in this way.
(1) The electromotive force of the induction of one circuit on
another is independent of the area of the section of the conductors
and of the material of which they are made.
For we can exchange any one of the circuits in the experiment
for another of a different section and material, but of the same form,
without altering the result.
* Annettes dc Chimie, xxxiv. p. G6 (1852), and Nuovo Cimento, ix. p. 345 (1859).
170 MAGNETO-ELECTRIC INDUCTION. [537-
(2) The induction of the circuit A on the circuit X is equal to
that of X upon A.
For if we put A in the galvanometer circuit, and X in the battery
circuit, the equilibrium of electromotive force is not disturbed.
(3) The induction is proportional to the inducing current.
For if we have ascertained that the induction of A on X is equal
to that of B on Y, and also to that of C on Z, we may make the
battery current first flow through A, and then divide itself in any
proportion between B and C. Then if we connect X reversed, Y
and Z direct, all in series, with the galvanometer, the electromotive
force in X will balance the sum of the electromotive forces in Y
(4) In pairs of circuits forming systems geometrically similar
the induction is proportional to their linear dimensions.
For if the three pairs of circuits above mentioned are all similar,
but if the linear dimension of the first pair is the sum of the
corresponding linear dimensions of the second and third pairs, then,
if A, B, and C are connected in series with the battery, and X
reversed, Y and Z also in series with the galvanometer, there will
be equilibrium.
(5) The electromotive force produced in a coil of n windings by
a current in a coil of m windings is proportional to the product mn.
537.] For experiments of the kind we have been considering the
galvanometer should be as sensitive as possible, and its needle as
light as possible, so as to give a sensible indication of a very
small transient current. The experiments on induction due to
motion require the needle to have a somewhat longer period of
vibration, so that there may be time to effect certain motions
of the conductors while the needle is not far from its position
of equilibrium. In the former experiments, the electromotive
forces in the galvanometer circuit were in equilibrium during
the whole time, so that no current passed through the galvano
meter coil. In those now to be described, the electromotive forces
act first in one direction and then in the other, so as to produce
in succession two currents in opposite directions through the gal
vanometer, and we have to shew that the impulses on the galvano
meter needle due to these successive currents are in certain cases
equal and opposite.
The theory of the application of the galvanometer to the
measurement of transient currents will be considered more at length
in Art. 748. At present it is sufficient for our purpose to ob-
538-J FELICl's EXPERIMENTS. 171
serve that as long- as the galvanometer needle is near its position
of equilibrium the deflecting force of the current is proportional
to the current itself, and if the whole time of action of the current
is small compared with the period of vibration of the needle, the
final velocity of the magnet will be proportional to the total
quantity of electricity in the current. Hence, if two currents pass
in rapid succession, conveying equal quantities of electricity in
opposite directions, the needle will be left without any final
velocity.
Thus, to shew that the induction-currents in the secondary circuit,
due to the closing and the breaking of the primary circuit, are
equal in total quantity but opposite in direction, we may arrange
the primary circuit in connexion with the battery, so that by
touching a key the current may be sent through the primary circuit,
or by removing the finger the contact may be broken at pleasure.
If the key is pressed down for some time, the galvanometer in
the secondary circuit indicates, at the time of making contact, a
transient current in the opposite direction to the primary current.
If contact be maintained, the induction current simply passes and
disappears. If we now break contact, another transient current
passes in the opposite direction through the secondary circuit,
and the galvanometer needle receives an impulse in the opposite
direction.
But if we make contact only for an instant, and then break
contact, the two induced currents pass through the galvanometer
in such rapid succession that the needle, when acted on by the first
current, has not time to move a sensible distance from its position
of equilibrium before it is stopped by the second, and, on account
of the exact equality between the quantities of these transient
currents, the needle is stopped dead.
If the needle is watched carefully, it appears to be jerked suddenly
from one position of rest to another position of rest very near
the first.
In this way we prove that the quantity of electricity in the
induction current, when contact is broken, is exactly equal and
opposite to that in the induction current when contact is made.
538.] Another application of this method is the following, which
is given by Felici in the second series of his Researches.
It is always possible to find many different positions of the
secondary coil I>, such that the making or the breaking of contact
in the primary coil A produces no induction current in 7?. The
172 MAGNETO-ELECTKIC INDUCTION. [539-
positions of the two coils are in such cases said to be conjugate to
each other.
Let BI and B2 be two of these positions. If the coil B be sud
denly moved from the position B± to the position J32, the algebraical
sum of the transient currents in the coil B is exactly zero, so
that the galvanometer needle is left at rest when the motion of B is
completed.
This is true in whatever way the coil B is moved from Bl to B2^
and also whether the current in the primary coil A be continued
constant, or made to vary during the motion.
Again, let B' be any other position of B not conjugate to A,
so that the making or breaking of contact in A produces an in
duction current when B is in the position B'.
Let the contact be made when B is in the conjugate position _Z?1?
there will be no induction current. Move B to B'> there will be
an induction current due to the motion, but if B is moved rapidly
to B', and the primary contact then broken, the induction current
due to breaking contact will exactly annul the effect of that due to
the motion, so that the galvanometer needle will be left at rest.
Hence the current due to the motion from a conjugate position
to any other position is equal and opposite to the current due to
breaking contact in the latter position.
Since the effect of making contact is equal and opposite to that
of breaking it, it follows that the effect of making contact when the
coil B is in any position B' is equal to that of bringing the coil
from any conjugate position Bl to B' while the current is flowing
through A.
If the change of the relative position of the coils is made by
moving the primary circuit instead of the secondary, the result is
found to be the same.
539.] It follows from these experiments that the total induction
current in B during the simultaneous motion of A from Al to A2J and
of B from Bl to B.2, while the current in A changes from ^ to y2,
depends only on the initial state AI} Bl, yl5 and the final state
A2, B2, y2, and not at all on the nature of the intermediate states
through which the system may pass.
Hence the value of the total induction current must be of the
form F(A2, B2, y2) - F(Alf £19 7l),
where F is a function of A, B, and y.
With respect to the form of this function, we know, by Art. 536,
that when there is no motion, and therefore Al = A2 and Bl = B2,
540.] ELECTROTONIC STATE. 173
the induction current is proportional to the primary current.
Hence y enters simply as a factor, the other factor being a func
tion of the form and position of the circuits A and J9.
We also know that the value of this function depends on the
relative and not on the absolute positions of A and B, so that
it must be capable of being1 expressed as a function of the distances
of the different elements of which the circuits are composed, and
of the angles which these elements make with each other.
Let M be this function, then the total induction current may be
written C {Ml7l-M2y.2},
where C is the conductivity of the secondary circuit, and M^ y1
are the original, and M2, y2 the final values of M and y.
These experiments, therefore, shew that the total current of
induction depends on the change which takes place in a certain
quantity, My, and that this change may arise either from variation
of the primary current y, or from any motion of the primary or
secondary circuit which alters M.
540.] The conception of such a quantity, on the changes of which,
and not on its absolute magnitude, the induction current depends,
occurred to Faraday at an early stage of his researches*. He
observed that the secondary circuit, when at rest in an electro
magnetic field which remains of constant intensity, does not shew
any electrical effect, whereas, if the same state of the field had been
suddenly produced, there would have been a current. Again, if the
primary circuit is removed from the field, or the magnetic forces
abolished, there is a current of the opposite kind. He therefore
recognised in the secondary circuit, when in the electromagnetic
field, a ' peculiar electrical condition of matter,' to which he gave
the name of the Electrotonic State. He afterwards found that he
could dispense with this idea by means of considerations founded on
the lines of magnetic force f, but even in his latest researches J,
he says, ( Again and again the idea of an electrotonic state § has
been forced upon my mind.'
The whole history of this idea in the mind of Faraday, as shewn
in his published researches, is well worthy of study. By a course
of experiments, guided by intense application of thought, but
without the aid of mathematical calculations, he was led to recog
nise the existence of something which we now know to be a mathe
matical quantity, and which may even be called the fundamental
* Exp. Res., series i. 60. % Ib., 3269.
t Ib., series ii. (242). § Ib., 60, 1114, 1661, 1729, 1733.
174 MAGNETO-ELECTRIC INDUCTION. [541*
quantity in the theory of electromagnetism. But as he was led
up to this conception by a purely experimental path, he ascribed
to it a physical existence, and supposed it to be a peculiar con
dition of matter, though he was ready to abandon this theory as
soon as he could explain the phenomena by any more familiar forms
of thought.
Other investigators were long afterwards led up to the same
idea by a purely mathematical path, but, so far as I know, none
of them recognised, in the refined mathematical idea of the potential
of two circuits, Faraday's bold hypothesis of an electrotonic state.
Those, therefore, who have approached this subject in the way
pointed out by those eminent investigators who first reduced its
laws to a mathematical form, have sometimes found it difficult
to appreciate the scientific accuracy of the statements of laws which
Faraday, in the first two series of his Researches, has given with
such wonderful completeness.
The scientific value of Faraday's conception of an electrotonic
state consists in its directing the mind to lay hold of a certain
quantity, on the changes of which the actual phenomena depend.
Without a much greater degree of development than Faraday gave
it, this conception does not easily lend itself to the explanation of the
phenomena. We shall return to this subject again in Art. 584.
541.] A method which, in Faraday's hands, was far more powerful
is that in which he makes use of those lines of magnetic force
which were always in his mind's eye when contemplating his
magnets or electric currents, and the delineation of which by
means of iron filings he rightly regarded * as a most valuable aid
to the experimentalist.
Faraday looked on these lines as expressing, not only by their
direction that of the magnetic force, but by their number and
concentration the intensity of that force, and in his later re
searches f he shews how to conceive of unit lines of force. I have
explained in various parts of this treatise the relation between the
properties which Faraday recognised in the lines of force and the
mathematical conditions of electric and magnetic forces, and how
Faraday's notion of unit lines and of the number of lines within
certain limits may be made mathematically precise. See Arts. 82,
404, 490.
In the first series of his Researches J he shews clearly how the
direction of the current in a conducting circuit, part of which is
* Exp. lies., 3234. t Ib., 3122. $ Ib., 114.
LINES OF MAGNETIC INDUCTION. 175
moveable, depends on the mode in which the moving1 part cuts
through the lines of magnetic force.
In the second series* he shews how the phenomena produced
by variation of the strength of a current or a magnet may be
explained, by supposing the system of lines of force to expand from
or contract towards the wire or magnet as its power rises or falls.
I am not certain with what degree of clearness he then held the
doctrine afterwards so distinctly laid down by him f, that the
moving conductor, as it cuts the lines of force, sums up the action
due to an area or section of the lines of force. This, however,
appears no new view of the case after the investigations of the
second series J have been taken into account.
The conception which Faraday had of the continuity of the lines
of force precludes the possibility of their suddenly starting into
existence in a place where there were none before. If, therefore,
the number of lines which pass through a conducting circuit is
made to vary, it can only be by the circuit moving across the lines
of force, or else by the lines of force moving across the circuit.
In either case a current is generated in the circuit.
The number of the lines of force which at any instant pass through
the circuit is mathematically equivalent to Faraday's earlier con
ception of the electrotonic state of that circuit, and it is represented
by the quantity My.
It is only since the definitions of electromotive force, Arts. 69,
274, and its measurement have been made more precise, that we
can enunciate completely the true law of magneto -electric induction
in the following terms : —
The total electromotive force acting round a circuit at any
instant is measured by the rate of decrease of the number of lines
of magnetic force which pass through it.
When integrated with respect to the time this statement be
comes : —
The time-integral of the total electromotive force acting round
any circuit, together with the number of lines of magnetic force
which pass through the circuit, is a constant quantity.
Instead of speaking of the number of lines of magnetic force, we
may speak of the magnetic induction through the circuit, or the
surface-integral of magnetic induction extended over any surface
bounded by the circuit.
* Exp. Res., 238. t Ib., 3082, 3087, 3113.
£ Ib., 217, &c.
176 MAGNETO-ELECTRIC INDUCTION. [542.
We shall return again to this method of Faraday. In the mean
time we must enumerate the theories of induction which are
founded on other considerations.
Lenz's Law.
542.] In 1834, Lenz* enunciated the following' remarkable
relation between the phenomena of the mechanical action of electric
currents, as defined by Ampere's formula, and the induction of
electric currents by the relative motion of conductors. An earlier
attempt at a statement of such a relation was given by Ritchie in
the Philosophical Magazine for January of the same year, but the
direction of the induced current was in every case stated wrongly.
Lenz's law is as follows. —
If a constant current flows in the primary circuit A, and if, by the
motion of A, or of the secondary circuit B, a current is induced in B, the
direction of this induced current wilt be such that, by its electromagnetic
action on A, it tends to oppose the relative motion of the circuits.
On this law J. Neumann f founded his mathematical theory of
induction, in which he established the mathematical laws of the
induced currents due to the motion of the primary or secondary
conductor. He shewed that the quantity M, which we have called
the potential of the one circuit on the other, is the same as the
electromagnetic potential of the one circuit on the other, which
we have already investigated in connexion with Ampere's formula.
We may regard J. Neumann, therefore, as having completed for
the induction of currents the mathematical treatment which Ampere
had applied to their mechanical action.
543.] A step of still greater scientific importance was soon after
made by Helmholtz in his Essay on the Conservation of Force J, and
by Sir W. Thomson §, working somewhat later, but independently
of Helmholtz. They shewed that the induction of electric currents
discovered by Faraday could be mathematically deduced from the
electromagnetic actions discovered by Orsted and Ampere by the
application of the principle of the Conservation of Energy.
Helmholtz takes the case of a conducting circuit of resistance R,
in which an electromotive force A, arising from a voltaic or thermo-
* Pogg., Ann. xxxi. 483 (1834).
t Berlin Acad., 1845 and 1847.
£ Kead before the Physical Society of Berlin, July 23, 1847. Translated in
Taylor's 'Scientific Memoirs,' part ii. p. 114.
§ Trans. Brit. Ass., 1848, and Phil. Mag., Dec. 1851. See also his paper on
'Transient Electric Currents,' Phil. Mag., 1853. .
543-1 HELMHOLTZ AND THOMSON. 177
electric arrangement, acts. The current in the circuit at any
instant is /. He supposes that a magnet is in motion in the
neighbourhood of the circuit, and that its potential with respect to
the conductor is F, so that, during any small interval of time dt, the
energy communicated to the magnet by the electromagnetic action
is
The work done in generating heat in the circuit is, by Joule's
law, Art. 242, I2 Belt, and the work spent by the electromotive
force A, in maintaining the current / during the time dt, is A Idt.
Hence, since the total work done must be equal to the work spent,
at
whence we find the intensity of the current
Now the value of A may be what we please. Let, therefore,
A = 0, and then 1
or, there will be a current due to the motion of the magnet, equal
dV
to that due to an electromotive force =- •
dt
The whole induced current during the motion of the magnet
from a place where its potential is V^ to a place where its potential
is Fo, is
and therefore the total current is independent of the velocity or
the path of the magnet, and depends only on its initial and final
positions.
In Helmholtz's original investigation he adopted a system of
units founded on the measurement of the heat generated in the
conductor by the current. Considering the unit of current as
arbitrary, the unit of resistance is that of a conductor in which this
unit current generates unit of heat in unit of time. The unit of
electromotive force in this system is that required to produce the
unit of current in the conductor of unit resistance. The adoption
of this system of units necessitates the introduction into the equa
tions of a quantity «, which is the mechanical equivalent of the
unit of heat. As we invariably adopt either the electrostatic or
VOL. II. N
178 MAGNETO-ELECTRIC INDUCTION. [544.
the electromagnetic system of units, this factor does not occur in
the equations here given.
544.] Helmholtz also deduces the current of induction when a
conducting circuit and a circuit carrying a constant current are
made to move relatively to one another.
Let Rlt R2 be the resistances, I19 I2 the currents, Alt A2 the
external electromotive forces, and V the potential of the one circuit
on the other due to unit current in each, then we have, as before,
4 /! + A, I2 = I^R, + L?R.> + /, 7, ~ •
If we suppose 7X to be the primary current, and 72 so much less
than /u that it does not by its induction produce any sensible
^
alteration in 715 so that we may put 7X = -— , then
a result which may be interpreted exactly as in the case of the
magnet.
If we suppose J2 to be the primary current, and I± to be very
much smaller than /2, we get for Ilt
A-I^
T AI L* dt
This shews that for equal currents the electromotive force of the
first circuit on the second is equal to that of the second on the first,
whatever be the forms of the circuits.
Helmholtz does not in this memoir discuss the case of induction
due to the strengthening or weakening of the primary current, or
the induction of a current on itself. Thomson * applied the same
principle to the determination of the mechanical value of a current,
and pointed out that when work is done by the mutual action of
two constant currents, their mechanical value is increased by the
same amount, so that the battery has to supply double that amount
of work, in addition to that required to maintain the currents
against the resistance of the circuits f.
545.] The introduction, by W. Weber, of a system of absolute
* Mechanical Theory of Electrolysis, Phil. Mag., Dec., 1851.
t Nichol's Cyclopaedia of Physical Science, ed. 1860, Article 'Magnetism, Dy
namical Relations of,' and Reprint, § 571.
545-1 WEBER. 179
units for the measurement of electrical quantities is one of the most
important steps in the progress of the science. Having already, in
conjunction with Gauss, placed the measurement of magnetic quan
tities in the first rank of methods of precision, Weher proceeded
in his Electrodynamic Measurements not only to lay down sound
principles for fixing the units to be employed, but to make de
terminations of particular electrical quantities in terms of these
units, with a degree of accuracy previously unattempted. Both the
electromagnetic and the electrostatic systems of units owe their
development and practical application to these researches.
Weber has also formed a general theory of electric action from
which he deduces both electrostatic and electromagnetic force, and
also the induction of electric currents. We shall consider this
theory, with some of its more recent developments, in a separate
chapter. See Art. 846.
N 2
CHAPTER IV.
ON THE INDUCTION OF A CURRENT ON ITSELF.
546.] FARADAY has devoted the ninth series of his Researches to
the investigation of a class of phenomena exhibited by the current
in a wire which forms the coil of an electromagnet.
Mr. Jenkin had observed that, although it is impossible to pro
duce a sensible shock by the direct action of a voltaic system
consisting of only one pair of plates, yet, if the current is made
to pass through the coil of an electromagnet, and if contact is
then broken between the extremities of two wires held one in each
hand, a smart shock will be felt. No such shock is felt on making
contact.
Faraday shewed that this and other phenomena, which he de
scribes, are due to the same inductive action which he had already
observed the current to exert on neighbouring conductors. In this
case, however, the inductive action is exerted on the same conductor
which carries the current, and it is so much the more powerful
as the wire itself is nearer to the different elements of the current
than any other wire can be.
547.] He observes, however *, that ' the first thought that arises
in the mind is that the electricity circulates with something like
momentum or inertia in the wire.' Indeed, when we consider one
particular wire only, the phenomena are exactly analogous to those
of a pipe full of water flowing in a continued stream. If while
the stream is flowing we suddenly close the end of the tube, the
momentum of the water produces a sudden pressure, which is much
greater than that due to the head of water, and may be sufficient
to burst the pipe.
If the water has the means of escaping through a narrow jet
* Exp. Res., 1077-
55O.] ELECTRIC INERTIA. 181
when the principal aperture is closed, it will be projected with a
velocity much greater than that due to the head of water, and
if it can escape through a valve into a chamber, it will do so,
even when the pressure in the chamber is greater than that due
to the head of water.
It is on this principle that the hydraulic ram is constructed,
by which a small quantity of water may be raised to a great height
by means of a large quantity flowing down from a much lower
level.
548.] These effects of the inertia of the fluid in the tube depend
solely on the quantity of fluid running through the tube, on its
length, and on its section in different parts of its length. They
do not depend on anything outside the tube, nor on the form into
which the tube may be bent, provided its length remains the
same.
In the case of the wire conveying a current this is not the case,
for if a long wire is doubled on itself the effect is very small, if
the two parts are separated from each other it is greater, if it
is coiled up into a helix it is still greater, and greatest of all if,
when so coiled, a piece of soft iron is placed inside the coil.
Again, if a second wire is coiled up with the first, but insulated
from it, then, if the second wire does not form a closed circuit,
the phenomena are as before, but if the second wire forms a closed
circuit, an induction current is formed in the second wire, and
the effects of self-induction in the first wire are retarded.
549.] These results shew clearly that, if the phenomena are due
to momentum, the momentum is certainly not that of the electricity
in the wire, because the same wire, conveying the same current,
exhibits effects which differ according to its form ; and even when
its form remains the same, the presence of other bodies, such as
a piece of iron or a closed metallic circuit, affects the result.
550.] It is difficult, however, for the mind which has once
recognised the analogy between the phenomena of self-induction
and those of the motion of material bodies, to abandon altogether
the help of this analogy, or to admit that it is entirely superficial
and misleading. The fundamental dynamical idea of matter, as
capable by its motion of becoming the recipient of momentum and
of energy, is so interwoven with our forms of thought that, when
ever we catch a glimpse of it in any part of nature, we feel that
a path is before us leading, sooner or later, to the complete under
standing of the subject.
182 SELF-INDUCTION. [551-
551.] In the case of the electric current, we find that, when the
electromotive force begins to act, it does not at once produce the
full current, but that the current rises gradually. What is the
electromotive force doing during the time that the opposing re
sistance is not able to balance it ? It is increasing the electric
current.
Now an ordinary force, acting on a body in the direction of its
motion, increases its momentum, and communicates to it kinetic
energy, or the power of doing work on account of its motion.
In like manner the unresisted part of the electromotive force has
been employed in increasing the electric current. Has the electric
current, when thus produced, either momentum or kinetic energy ?
We have already shewn that it has something very like mo
mentum, that it resists being suddenly stopped, and that it can
exert, for a short time, a great electromotive force.
But a conducting circuit in which a current has been set up
has the power of doing work in virtue of this current, and this
power cannot be said to be something very like energy, for it
is really and truly energy.
Thus, if the current be left to itself, it will continue to circulate
till it is stopped by the resistance of the circuit. Before it is
stopped, however, it will have generated a certain quantity of
heat, and the amount of this heat in dynamical measure is equal
to the energy originally existing in the current.
Again, when the current is left to itself, it may be made to
do mechanical work by moving magnets, and the inductive effect
of these motions will, by Lenz's law, stop the current sooner than
the resistance of the circuit alone would have stopped it. In this
way part of the energy of the current may be transformed into
mechanical work instead of heat.
552.] It appears, therefore, that a system containing an electric
current is a seat of energy of some kind ; and since we can form
no conception of an electric current except as a kinetic pheno
menon *, its energy must be kinetic energy, that is to say, the
energy which a moving body has in virtue of its motion.
We have already shewn that the electricity in the wire cannot
be considered as the moving body in which we are to find this
energy, for the energy of a moving body does not depend on
anything external to itself, whereas the presence of other bodies
near the current alters its energy.
* Faraday, Eocp. Res. (283.)
552.] ELECTROKINETIC ENEKGY. 183
We are therefore led to enquire whether there may not be some
motion going1 on in the space outside the wire, which is not occupied
by the electric current, but in which the electromagnetic effects of
the current are manifested.
I shall not at present enter on the reasons for looking in one
place rather than another for such motions, or for regarding these
motions as of one kind rather than another.
What I propose now to do is to examine the consequences of
the assumption that the phenomena of the electric current are those
of a moving system, the motion being communicated from one part
of the system to another by forces, the nature and laws of which
we do not yet even attempt to define, because we can eliminate
these forces from the equations of motion by the method given
by Lagrange for any connected system.
In the next five chapters of this treatise I propose to deduce
the main structure of the theory of electricity from a dynamical
hypothesis of this kind, instead of following the path which has
led Weber and other investigators to many remarkable discoveries
and experiments, and to conceptions, some of which are as beautiful
as they are bold. I have chosen this method because I wish to
shew that there are other ways of viewing the phenomena which
appear to me more satisfactory, and at the same time are more
consistent with the methods followed in the preceding parts of this
book than those which proceed on the hypothesis of direct action
at a distance.
CHAPTER V.
ON THE EQUATIONS OF MOTION OF A CONNECTED SYSTEM.
553.] IN the fourth section of the second part of his Mecanique
Analytique, Lagrange has given a method of reducing the ordinary
dynamical equations of the motion of the parts of a connected
system to a number equal to that of the degrees of freedom of
the system.
The equations of motion of a connected system have been given
in a different form by Hamilton, and have led to a great extension
of the higher part of pure dynamics *.
As we shall find it necessary, in our endeavours to bring electrical
phenomena within the province of dynamics, to have our dynamical
ideas in a state fit for direct application to physical questions, we
shall devote this chapter to an exposition of these dynamical ideas
from a physical point of view.
554.] The aim of Lagrange was to bring dynamics under the
power of the calculus. He began by expressing the elementary
dynamical relations in terms of the corresponding relations of pure
algebraical quantities, and from the equations thus obtained he
deduced his final equations by a purely algebraical process. Certain
quantities (expressing the reactions between the parts of the system
called into play by its physical connexions) appear in the equations
of motion of the component parts of the system, and Lagrange's
investigation, as seen from a mathematical point of view, is a
method of eliminating these quantities from the final equations.
In following the steps of this elimination the mind is exercised
in calculation, and should therefore be kept free from the intrusion
of dynamical ideas. Our aim, on the other hand, is to cultivate
* See Professor Cayley's ' Report on Theoretical Dynamics,' British Association,
3 857 ; and Thomson and Tait's Natural Philosophy.
555-] GENERALIZED COORDINATES. 185
our dynamical ideas. We therefore avail ourselves of the labours
of the mathematicians, and retranslate their results from the lan
guage of the calculus into the language of dynamics, so that our
words may call up the mental image, not of some algebraical
process, but of some property of moving bodies.
The language of dynamics has been considerably extended by
those who have expounded in popular terms the doctrine of the
Conservation of Energy, and it will be seen that much of the
following statement is suggested by the investigation in Thomson
and Tait^s Natural Philosophy, especially the method of beginning
with the theory of impulsive forces.
I have applied this method so as to avoid the explicit con
sideration of the motion of any part of the system except the
coordinates or variables, on which the motion of the whole depends.
It is doubtless important that the student should be able to trace
the connexion of the motion of each part of the system with that
of the variables, but it is by no means necessary to do this in
the process of obtaining the final equations, which are independent
of the particular form of these connexions.
The Variables.
555.] The number of degrees of freedom of a system is the
number of data which must be given in order completely to
determine its position. Different forms may be given to these
data, but their number depends on the nature of the system itself,
and cannot be altered.
To fix our ideas we may conceive the system connected by means
of suitable mechanism with a number of moveable pieces, each
capable of motion along a straight line, and of no other kind of
motion. The imaginary mechanism which connects each of these
pieces with the system must be conceived to be free from friction,
destitute of inertia, and incapable of being strained by the action
of the applied forces. The use of this mechanism is merely to
assist the imagination in ascribing position, velocity, and momentum
to what appear, in Lagrange's investigation, as pure algebraical
quantities.
Let q denote the position of one of the moveable pieces as defined
by its distance from a fixed point in its line of motion. We shall
distinguish the values of q corresponding to the different pieces
by the suffixes u 2, &c. When we are dealing with a set of
quantities belonging to one piece only we may omit the suffix.
186 KINETICS. [556.
When the values of all the variables (q) are given, the position
of each of the moveable pieces is known, and, in virtue of the
imaginary mechanism, the configuration of the entire system is
determined.
The Velocities.
556.] During the motion of the system the configuration changes
in some definite manner, and since the configuration at each instant
is fully defined by the values of the variables (q), the velocity of
every part of the system, as well as its configuration, will be com
pletely defined if we know the values of the variables (q), together
with their velocities (-— , or, according to Newton's notation, q) •
The Forces.
557.] By a proper regulation of the motion of the variables, any
motion of the system, consistent with the nature of the connexions,
may be produced. In order to produce this motion by moving
the variable pieces, forces must be applied to these pieces.
We shall denote the force which must be applied to any variable
qr by Fr. The system of forces (F) is mechanically equivalent (in
virtue of the connexions of the system) to the system of forces,
whatever it may be, which really produces the motion.
The Momenta.
558.] When a body moves in such a way that its configuration,
with respect to the force which acts on it, remains always the same,
(as, for instance, in the case of a force acting on a single particle in
the line of its motion,) the moving force is measured by the rate
of increase of .the momentum. If F is the moving force, and p the
momentum,
whence p = / Fdt.
The time-integral of a force is called the Impulse of the force ;
so that we may assert that the momentum is the impulse of the
force which would bring the body from a state of rest into the given
state of motion.
In the case of a connected system in motion, the configuration is
continually changing at a rate depending on the velocities (q\ so
559-] IMPULSE AND MOMENTUM. 187
that we can no longer assume that the momentum is the time-
intesral of the force which acts on it.
o
But the increment bq of any variable cannot be greater than
qbt, where 8^ is the time during which the increment takes place,
and q is the greatest value of the velocity during that time. In the
case of a system moving from rest under the action of forces always
in the same direction, this is evidently the final velocity.
If the final velocity and configuration of the system are given,
we may conceive the velocity to be communicated to the system
in a very small time §t, the original configuration differing from
the final configuration by quantities bqlt §£2, &c., which are less
than q^btj ^25^, &c., respectively.
The smaller we suppose the increment of time 8£, the greater
must be the impressed forces, but the time-integral, or impulse,
of each force will remain finite. The limiting value of the impulse,
when the time is diminished and ultimately vanishes, is defined
as the instantaneous impulse, and the momentum p, corresponding
to any variable q, is defined as the impulse corresponding to that
variable, when the system is brought instantaneously from a state
of rest into the given state of motion.
This conception, that the momenta are capable of being produced
by instantaneous impulses on the system at rest, is introduced only
as a method of defining the magnitude of the momenta, for the
momenta of the system depend only on the instantaneous state
of motion of the system, and not on the process by which that state
was produced.
In a connected system the momentum corresponding to any
variable is in general a linear function of the velocities of all the
variables, instead of being, as in the dynamics of a particle, simply
proportional to the velocity.
The impulses required to change the velocities of the system
suddenly from yl9 q.2, &c. to £/, q2', &c, are evidently equal to
Pi — p\, Pz — J°2> ^ne cbaBgcs of momentum of the several variables.
Work done by a Small Impulse.
559.] The work done by the force Fl during the impulse is the
space-integral of the force, or
W
=j
188 KINETICS. [560.
If fa is the greatest and q" the least value of tlie velocity q-^
during the action of the force, W must be less than
2i< Fdt
or
and greater than q"\Fdt or q\(p\—p\)>
If we now suppose the impulse / Fdt to be diminished without
limit, the values of q{ and q" will approach and ultimately coincide
with that of qlt and we may write p{—p^ = §pi, so that the work
done is ultimately 7ir
or, the work done by a very small impulse is ultimately the product
of the impulse and the velocity.
Increment of the Kinetic Energy.
560.] When work is done in setting a conservative system in
motion, energy is communicated to it, and the system becomes
capable of doing an equal amount of work against resistances
before it is reduced to rest.
The energy which a system possesses in virtue of its motion
is called its Kinetic Energy, and is communicated to it in the form
of the work done by the forces which set it in motion.
If T be the kinetic energy of the system, and if it becomes
T 4- 8 T} on account of the action of an infinitesimal impulse whose
components are 8^15 5j02, &c., the increment 8 T must be the sum
of the quantities of work done by the components of the impulse,
or in symbols, IT = &*& + js 8A + &c.,
= 2&8j»). (1)
The instantaneous state of the system is completely defined if
the variables and the momenta are given. Hence the kinetic
energy, which depends on the instantaneous state of the system,
can be expressed in terms of the variables (q), and the momenta (/>).
This is the mode of expressing T introduced by Hamilton. When
T is expressed in this way we shall distinguish it by the suffix p)
thus, Tp.
The complete variation of Tp is
^=2^+Ss?. (2)
561.] HAMILTON'S EQUATIONS. 189
The last term may be written
which diminishes with 8£, and ultimately vanishes with it when the
impulse becomes instantaneous.
Hence, equating- the coefficients of bp in equations (1) and (2),
we obtain . = ^ (s)
or, the velocity corresponding to the variable q is the differential
coefficient of Tp with respect to the corresponding momentum p.
We have arrived at this result by the consideration of impulsive
forces. By this method we have avoided the consideration of the
change of configuration during the action of the forces. But the
instantaneous state of the system is in all respects the same, whether
the system was brought from a state of rest to the given state
of motion by the transient application of impulsive forces, or
whether it arrived at that state in any manner, however gradual.
In other words, the variables, and the corresponding velocities
and momenta, depend on the actual state of motion of the system
at the given instant, and not on its previous history.
Hence, the equation (3) is equally valid, whether the state of
motion of the system is supposed due to impulsive forces, or to
forces acting in any manner whatever.
We may now therefore dismiss the consideration of impulsive
forces, together with the limitations imposed on their time of
action, and on the changes of configuration during their action.
Hamilton's Equations of Motion.
561.] We have already shewn that
dT
(4)
Let the system move in any arbitrary way, subject to the con
ditions imposed by its connexions, then the variations of p and q are
(5)
190 KINETICS. [562.
and the complete variation of Tp is
But the increment of the kinetic energy arises from the work
done by the impressed forces, or
IT, = 2 (Fig). (8)
In these two expressions the variations bq are all independent of
each other, so that we are entitled to equate the coefficients of each
of them in the two expressions (7) and (8). We thus obtain
where the momentum^ and the force Fr belong to the variable qr.
There are as many equations of this form as there are variables.
These equations were given by Hamilton They shew that the
force corresponding to any variable is the sum of two parts. The
first part is the rate of increase of the momentum of that variable
with respect to the time. The second part is the rate of increase
of the kinetic energy per unit of increment of the variable, the
other variables and all the momenta being constant.
The Kinetic Energy expressed in Terms of the Momenta and
Velocities.
562.] Let pl9 p2, &c. be the momenta, and ql} q2, &c. the
velocities at a given instant, and let px, p2, &c., qx, q2, &c. be
another system of momenta and velocities, such that
Pi = *Pi> 4i = »0n &c- (10)
It is manifest that the systems p, q will be consistent with each
other if the systems p, q are so.
Now let n vary by bn. The work done by the force Fl is
Fi*h = 4i8Pi = Jiftntn. (11)
Let n increase from 0 to 1, then the system is brought from
a state of rest into the state of motion (qp), and the whole work
expended in producing this motion is
-)/
But
ri
/ ndn = \,
Jn
564.] LAGRANGE'S EQUATIONS. 191
and the work spent in producing the motion is equivalent to the
kinetic energy. Hence
TP*= iC^ift + ^fc + fcC'). (13)
where Tp$ denotes the kinetic energy expressed in terms of the
momenta and velocities. The variables ql, q% , &c. do not enter into
this expression.
The kinetic energy is therefore half the sum of the products of
the momenta into their corresponding velocities.
When the kinetic energy is expressed in this way we shall denote
it by the symbol Tp^ . It is a function of the momenta and velo
cities only, and does not involve the variables themselves.
563.] There is a third method of expressing the kinetic energy,
which is generally, indeed, regarded as the fundamental one. By
solving the equations (3) we may express the momenta in terms
of the velocities, and then, introducing these values in (13), we shall
have an expression for T involving only the velocities and the
variables. When T is expressed in .this form we shall indicate it
by the symbol T^ . This is the form in which the kinetic energy is
expressed in the equations of Lagrange.
564.] It is manifest that, since Tp, T$9 and Tp^ are three different
expressions for the same thing,
Tp+Tt-2Tp(l = 0,
or Tp + Tt-Piii-toto-ke. = °- (14)
Hence, if all the quantities jo, q, and q vary,
The variations 8jt? are not independent of the variations bq and
bq, so that we cannot at once assert that the coefficient of each
variation in this equation is zero. But we know, from equations
(3)'that g-ft = o,fa, do)
so that the terms involving the variations bp vanish of themselves.
The remaining variations bq and bq are now all independent,
so that we find, by equating to zero the coefficients of bqlt &c ,
192 KINETICS. [565.
or, the components of momentum are the differential coefficients of T^
with respect to the corresponding velocities.
Again, by equating to zero the coefficients of 8^15 &c.,
^+^ = 0; (.8)
dch d^
or, the differential coefficient of the kinetic energy with respect to any
variable ql is equal in magnitude but opposite in sign when T is
expressed as a function of the velocities instead of as a function of
the momenta.
In virtue of equation (18) we may write the equation of motion (9),
p djiw,
dt dql
p i'VtW (20)
at dql dql
which is the form in which the equations of motion were given by
Lagrange.
565.] In the preceding investigation we have avoided the con
sideration of the form of the function which expresses the kinetic
energy in terms either of the velocities or of the momenta. The
only explicit form which we have assigned to it is
TP* = 4 (PiJi + J»2? + &c.), (21)
in which it is expressed as half the sum of the products of the
momenta each into its corresponding velocity.
We may express the velocities in terms of the differential co
efficients of Tp with respect to the momenta, as in equation (3),
This shews that Tp is a homogeneous function of the second
degree of the momenta pl} p2, &c.
We may also express the momenta in terms of T$ , and we find
*«-*&§+*§ + *"•) <23>
which shews that T$ is a homogeneous function of the second degree
with respect to the velocities <?15 q2, &c.
If we write
Pn for ^, P12 for ^-±« &c.
and Qn for - ? , Q12 for -^ — /- , &c. ;
567.] MOMENTS AND PRODUCTS OF INERTIA. 193
then, since both T(j and Tp are functions of the second degree of
q and of p respectively, both the P's and the Q's will be functions
of the variables q only, and independent of the velocities and the
momenta. We thus obtain the expressions for I\
2 TI = Pn tf + 2P12 q, q2 + &c., (24)
2Tp= QuPi2 + 2 Qi2PiP2 + &c- (25)
The momenta are expressed in terms of the velocities by the
linear equations ^ = pn ^ + P12 ^ + &c., (26)
and the velocities are expressed in terms of the momenta by the
linear equations ^ = Qn p± + Q12p2 + &c. (27)
In treatises on the dynamics of a rigid body, the coefficients
corresponding to Pn, in which the suffixes are the same, are called
Moments of Inertia, and those corresponding to P12, in which
the suffixes are different, are called Products of Inertia. We may
extend these names to the more general problem which is now
before us, in which these quantities are not, as in the case of a
rigid body, absolute constants, but are functions of the variables
In like manner we may call the coefficients of the form Qn
Moments of Mobility, and those of the form Q12, Products of
Mobility. It is not often, however, that we shall have occasion
to speak of the coefficients of mobility.
566.] The kinetic energy of the system is a quantity essentially
positive or zero. Hence, whether it be expressed in terms of the
velocities, or in terms of the momenta, the coefficients must be
such that no real values of the variables can make T negative.
We thus obtain a set of necessary conditions which the values of
the coefficients P must satisfy.
The quantities Pn, P22, &c., and all determinants of the sym
metrical form
P P P
12 22 '
p p p
•*• 13 •*• 23 •* q
which can be formed from the system of coefficients must be positive
or zero. The number of such conditions for n variables is 2n— 1.
The coefficients Q are subject to conditions of the same kind.
567.] In this outline of the fundamental principles of the dy
namics of a connected system, we have kept out of view the
mechanism by which the parts of the system are connected. We
VOL. n. o
194 KINETICS. [567.
have not even written down a set of equations to indicate how
the motion of any part of the system depends on the variation
of the variables. We have confined our attention to the variables,
their velocities and momenta, and the forces which act on the
pieces representing- the variables. Our only assumptions are, that
the connexions of the system are such that the time is not explicitly
contained in the equations of condition, and that the principle of
the conservation of energy is applicable to the system.
Such a description of the methods of pure dynamics is not un
necessary, because Lag-range and most of his followers, to whom
we are indebted for these methods, have in general confined them
selves to a demonstration of them, and, in order to devote their
attention to the symbols before them, they have endeavoured to
banish all ideas except those of pure quantity, so as not only to
dispense with diagrams, but even to get rid of the ideas of velocity,
momentum, and energy, after they have been once for all sup
planted by symbols in the original equations. In order to be able
to refer to the results of this analysis in ordinary dynamical lan
guage, we have endeavoured to retranslate the principal equations
of the method into language which may be intelligible without the
use of symbols.
As the development of the ideas and methods of pure mathe
matics has rendered it possible, by forming a mathematical theory
of dynamics, to bring to light many truths which could not have
been discovered without mathematical training, so, if we are to
form dynamical theories of other sciences, we must have our minds
imbued with these dynamical truths as well as with mathematical
methods.
In forming the ideas and words relating to any science, which,
like electricity, deals with forces and their effects, we must keep
constantly in mind the ideas appropriate to the fundamental science
of dynamics, so that we may, during the first development of the
science, avoid inconsistency with what is already established, and
also that when our views become clearer, the language we have
adopted may be a help to us and not a hindrance.
CHAPTER VI.
DYNAMICAL THEORY OF ELECTROMAGNETISM.
568.] WE have shewn, in Art. 552, that, when an electric current
exists in a conducting circuit, it has a capacity for doing a certain
amount of mechanical work, and this independently of any external
electromotive force maintaining the current. Now capacity for
performing work is nothing else than energy, in whatever way
it arises, and all energy is the same in kind, however it may differ
in form. The energy of an electric current is either of that form
which consists in the actual motion of matter, or of that which
consists in the capacity for being set in motion, arising from forces
acting between bodies placed in certain positions relative to each
other.
The first kind of energy, that of motion, is called Kinetic energy,
and when once understood it appears so fundamental a fact of
nature that we can hardly conceive the possibility of resolving
it into anything else. The second kind of energy, that depending
on position, is called Potential energy, and is due to the action
of what we call forces, that is to say, tendencies towards change
of relative position. With respect to these forces, though we may
accept their existence as a demonstrated fact, yet we always feel
that every explanation of the mechanism by which bodies are set
in motion forms a real addition to our knowledge.
569.] The electric current cannot be conceived except as a kinetic
phenomenon. Even Faraday, who constantly endeavoured to
emancipate his mind from the influence of those suggestions which
the words ' electric current' and ' electric fluid' are too apt to carry
with them, speaks of the electric current as ' something progressive,
and not a mere arrangement ' *.
* Exp. Res., 283.
O 2
196 ELECTROKINETICS. \_S7°-
The effects of the current, such as electrolysis, and the transfer
of electrification from one body to another, are all progressive
actions which require time for their accomplishment, and are there
fore of the nature of motions.
As to the velocity of the current, we have shewn that we know
nothing about it, it may be the tenth of an inch in an hour, or
a hundred thousand miles in a second *. So far are we from
knowing its absolute value in any case, that we do not even know
whether what we call the positive direction is the actual direction
of the motion or the reverse.
But all that we assume here is that the electric current involves
motion of some kind. That which is the cause of electric currents
has been called Electromotive Force. This name has long been
used with great advantage, and has never led to any inconsistency
in the language of science. Electromotive force is always to be
understood to act on electricity only, not on the bodies in which
the electricity resides. It is never to be confounded with ordinary
mechanical force, which acts on bodies only, not on the electricity
in them. If we ever come to know the formal relation between
electricity and ordinary matter, we shall probably also know the
relation between electromotive force and ordinary force.
570.] When ordinary force acts on a body, and when the body
yields to the force, the work done by the force is measured by the
product of the force into the amount by which the body yields.
Thus, in the case of water forced through a pipe, the work done
at any section is measured by the fluid pressure at the section
multiplied into the quantity of water which crosses the section.
In the same way the work done by an electromotive force is
measured by the product of the electromotive force into the quantity
of electricity which crosses a section of the conductor under the
action of the electromotive force.
The work done by an electromotive force is of exactly the same
kind as the work done by an ordinary force, and both are measured
by the same standards or units.
Part of the work done by an electromotive force acting on a
conducting circuit is spent in overcoming the resistance of the
circuit, and this part of the work is thereby converted into heat.
Another part of the work is spent in producing the electromag
netic phenomena observed by Ampere, in which conductors are
made to move by electromagnetic forces. The rest of the work
* Exp. Res., 1648.
KINETIC ENEEGY. 197
is spent in increasing the kinetic energy of the current, and the
effects of this part of the action are shewn in the phenomena of the
induction of currents observed by Faraday.
We therefore know enough about electric currents to recognise,
in a system of material conductors carrying currents, a dynamical
system which is the seat of energy, part of which may be kinetic
and part potential.
The nature of the connexions of the parts of this system is
unknown to us, but as we have dynamical methods of investigation
which do not require a knowledge of the mechanism of the system,
we shall apply them to this case.
We shall first examine the consequences of assuming the most
general form for the function which expresses the kinetic energy of
the system.
571.] Let the system consist of a number of conducting circuits,
the form and position of which are determined by the values of
a system of variables #15 x9) &c., the number of which is equal
to the number of degrees of freedom of the system.
If the whole kinetic energy of the system were that due to the
motion of these conductors, it would be expressed in the form
T = i (#! ffj a?!2 -f &c. + (^ a?2) ^ x2 -f &c.,
where the symbols (^15 a:lf &c.) denote the quantities which we have
called moments of inertia, and (#1} sc29 &c.) denote the products of
inertia.
If X' is the impressed force, tending to increase the coordinate x,
which is required to produce the actual motion, then, by Lagrange's
equation, d dT dT _
dt dx dx ~
When T denotes the energy due to the visible motion only, we
shall indicate it by the suffix TO, thus, Tm.
But in a system of conductors carrying electric currents, part of
the kinetic energy is due to the existence of these currents. Let
the motion of the electricity, and of anything whose motion is
governed by that of the electricity, be determined by another set
of coordinates y^ y2, &c., then T will be a homogeneous function
of squares and products of all the velocities of the two sets of
coordinates. We may therefore divide T into three portions, in the
first of which, Tm, the velocities of the coordinates x only occur,
while in the second, Te, the velocities of the coordinates y only
occur, and in the third, Tme, each term contains the product of the
velocities of two coordinates of which one is as and the other y.
198 ELECTROKINETICS.
We have therefore T — T _L T -4- T
•* — -Lm-T--Le^r Lme)
where Tm = | (^ ^) ^2 -f &c. + (^ #2) ^ #2 + &c->
572.] In the general dynamical theory, the coefficients of every
term may be functions of all the coordinates, both x and y. In
the case of electric currents, however, it is easy to see that the
coordinates of the class y do not enter into the coefficients.
For, if all the electric currents are maintained constant, and the
conductors at rest, the whole state of the field will remain constant.
But in this case the coordinates y are variable, though the velocities
y are constant. Hence the coordinates y cannot enter into the
expression for T, or into any other expression of what actually takes
place.
Besides this, in virtue of the equation of continuity, if the con
ductors are of the nature of linear circuits, only one variable is
required to express the strength of the current in each conductor.
Let the velocities y^yz, &c. represent the strengths of the currents
in the several conductors.
All this would be true, if, instead of electric currents, we had
currents of an incompressible fluid running in flexible tubes. In
this case the velocities of these currents would enter into the
expression for T, but the coefficients would depend only on the
variables x, which determine the form and position of the tubes.
In the case of the fluid, the motion of the fluid in one tube does
not directly affect that of any other tube, or of the fluid in it.
Hence, in the value of T6, only the squares of the velocities y, and
not their products, occur, and in T^ any velocity y is associated
only with those velocities of the form x which belong to its own
tube.
In the case of electrical currents we know that this restriction
does not hold, for the currents in different circuits act on each other.
Hence we must admit the existence of terms involving products
of the form y±y^ and this involves the existence of something in
motion, whose motion depends on the strength of both electric
currents y^ and y2. This moving matter, whatever it is, is not
confined to the interior of the conductors carrying the two currents,
but probably extends throughout the whole space surrounding them.
573.] Let us next consider the form which Lagrange's equations
of motion assume in this case. Let X' be the impressed force
573-] ELECTROMAGNETIC FORCE. 199
corresponding- to the coordinate a?, one of those which determine
the form and position of the conducting- circuits. This is a force
in the ordinary sense, a tendency towards change of position. It
is given by the equation
x/_ cl_dT^_dT_
dt dx dx
We may consider this force as the sum of three parts, corre
sponding to the three parts into which we divided the kinetic
energy of the system, and we may distinguish them by the same
suffixes. Thus -%•' _ T
The part X'm is that which depends on ordinary dynamical con
siderations, and we need not attend to it.
Since T0 does not contain x, the first term of the expression
for X'e is zero, and its value is reduced to
J' dT*
«~ ~ dx '
This is the expression for the mechanical force which must be
applied to a conductor to balance the electromagnetic force, and it
asserts that it is measured by the rate of diminution of the purely
electrokinetic energy due to the variation of the coordinate x. The
electromagnetic force, Xe, which brings this external mechanical
force into play, is equal and opposite to it, and is therefore measured
by the rate of increase of the electrokinetic energy corresponding
to an increase of the coordinate x. The value of Xe, since it depends
on squares and products of the currents, remains the same if we
reverse the directions of all the currents.
The third part of X' is
d dTme dT^
_
me~ dt dx dx
The quantity Tme contains only products of the form xy, so that
dT
me is a linear function of the strengths of the currents i/. The
first term, therefore, depends on the rate of variation of the
strengths of the currents, and indicates a mechanical force on
the conductor, which is zero when the currents are constant, and
which is positive or negative according as the currents are in
creasing or decreasing in strength.
The second term depends, not on the variation of the currents,
but on their actual strength. As it is a linear function with
respect to these currents, it changes sign when the currents change
200
ELECTROKINETICS.
[574.
sign. Since every term involves a velocity x, it is zero when the
conductors are at rest.
We may therefore investigate these terms separately. If the
conductors are at rest, we have only the first term to deal with.
If the currents are constant, we have only the second.
574.] As it is of great importance to determine whether any
part of the kinetic energy is of the form Tme, consisting of products
of ordinary velocities and strengths of electric currents, it is de
sirable that experiments should be made on this subject with great
care.
The determination of the forces acting on bodies in rapid motion
is difficult. Let us therefore attend to the first term, which depends
on the variation of the strength of the current.
If any part of the kinetic energy depends on the product of
an ordinary velocity and the strength of a
current, it will probably be most easily ob
served when the velocity and the current are
in the same or in opposite directions. We
therefore take a circular coil of a great many
windings, and suspend it by a fine vertical wire,
so that its windings are horizontal, and the
coil is capable of rotating about a vertical axis,
either in the same direction as the current in
the coil, or in the opposite direction.
We shall suppose the current to be conveyed
into the coil by means of the suspending wire,
and, after passing round the windings, to com
plete its circuit by passing downwards through
a wire in the same line with the suspending
wire and dipping into a cup of mercury.
Since the action of the horizontal component
pj 33 of terrestrial magnetism would tend to turn
this coil round a horizontal axis when the
current flows through it, we shall suppose that the horizontal com
ponent of terrestrial magnetism is exactly neutralized by means
of fixed magnets, or that the experiment is made at the magnetic
pole. A vertical mirror is attached to the coil to detect any motion
in azimuth.
Now let a current be made to pass through the coil in the
direction N.E.S.W. If electricity were a fluid like water, flowing
along the wire, then, at the moment of starting the current, and as
574-1 HAS AN" ELECTRIC CURRENT TRUE MOMENTUM.7? 201
long as its velocity is increasing, a force would require to be
supplied to produce the angular momentum of the fluid in passing
round the coil, and as this must be supplied by the elasticity of
the suspending wire, the coil would at first rotate in the opposite
direction or W.S.E.N., and this would be detected by means of
the mirror. On stopping the current there would be another
movement of the mirror, this time in the same direction as that
of the current.
No phenomenon of this kind has yet been observed. Such an
action, if it existed, might be easily distinguished from the already
known actions of the current by the following peculiarities.
(1) It would occur only when the strength of the current varies,
as when contact is made or broken, and not when the current is
constant.
All the known mechanical actions of the current depend on the
strength of the currents, and not on the rate of variation. The
electromotive action in the case of induced currents cannot be
confounded with this electromagnetic action.
(2) The direction of this action would be reversed when that
of all the currents in the field is reversed.
All the known mechanical actions of the current remain the same
when all the currents are reversed, since they depend on squares
and products of these currents.
If any action of this kind were discovered, we should be able
to regard one of the so-called kinds of electricity, either the positive
or the negative kind, as a real substance, and we should be able
to describe the electric current as a true motion of this substance
in a particular direction. In fact, if electrical motions were in any
way comparable with the motions of ordinary matter, terms of the
form Tme would exist, and their existence would be manifested by
the mechanical force Xm, .
According to Fechner's hypothesis, that an electric current con
sists of two equal currents of positive and negative electricity,
flowing in opposite directions through the same conductor, the
terms of the second class Tme would vanish, each term belonging
to the positive current being accompanied by an equal term of
opposite sign belonging to the negative current, and the phe
nomena depending on these terms would have no existence.
It appears to me, however, that while we derive great advantage
from the recognition of the many analogies between the electric
current and a current of a material fluid, we must carefully avoid
202
ELECTROKINETICS.
[575-
making any assumption not warranted by experimental evidence,
and that there is, as yet, no experimental evidence to shew whether
the electric current is really a current of a material substance, or
a double current, or whether its velocity is great or small as mea
sured in feet per second.
A knowledge of these things would amount to at least the begin
nings of a complete dynamical theory of electricity, in which we
should regard electrical action, not, as in this treatise, as a phe
nomenon due to an unknown cause, subject only to the general
laws of dynamics, but as the result of known motions of known
portions of matter, in which not only the total effects and final
results, but the whole intermediate mechanism and details of the
motion, are taken as the objects of study.
575.] The experimental investigation of the second term of Xme,
dT
namely -- r — , is more difficult, as it involves the observation of
ax
the effect of forces on a body in rapid motion.
Fig. 34.
The apparatus shewn in Fig. 34, which I had constructed in
1861, is intended to test the existence of a force of this kind.
575-] EXPERIMENT OF ROTATION. 203
The electromagnet A is capable of rotating' about the horizontal
axis BB', within a ring which itself revolves about a vertical
axis.
Let A, J5, C be the moments of inertia of the electromagnet
about the axis of the coil, the horizontal axis BB' , and a third axis
CC' respectively.
Let 6 be the angle which CG' makes with the vertical, </> the
azimuth of the axis BB', and \f/ a variable on which the motion of
electricity in the coil depends.
Then the kinetic energy of the electromagnet may be written
2 T = A & sin2 0 + B 62 + <7<j>2 cos2 0 + E (<£ sin 6 + ^)2,
where E is a quantity which may be called the moment of inertia
of the electricity in the coil.
If 0 is the moment of the impressed force tending to increase 0,
we have, by the equations of dynamics,
d2Q • . . .
0 = B -r^— {(A— C)02sm0cos0 + ^(£cos0((/>sm<9 + \//)}.
(It
By making % the impressed force tending to increase \j/t equal
to zero, we obtain
<£ sin 0 -f x//- = y,
a constant, which we may consider as representing the strength of
the current in the coil.
If C is somewhat greater than A, 0 will be zero, and the equi
librium about the axis BB' will be stable when
Ey
sin 0 = - — — r •
This value of 0 depends on that of y, the electric current, and
is positive or negative according to the direction of the current.
The current is passed through the coil by its bearings at B
and B', which are connected with the battery by means of springs
rubbing on metal rings placed on the vertical axis.
To determine the value of 0, a disk of paper is placed at C,
divided by a diameter parallel to BB' into two parts, one of which
is painted red and the other green.
When the instrument is in motion a red circle is seen at C
when 0 is positive, the radius of which indicates roughly the value
of 0. When 0 is negative, a green circle is seen at C.
By means of nuts working on screws attached to the electro
magnet, the axis CC' is adjusted to be a principal axis having
its moment of inertia just exceeding that round the axis A, so as
204 ELECTROKINETICS. [5?6.
to make the instrument very sensible to the action of the force
if it exists.
The chief difficulty in the experiments arose from the disturbing
action of the earth's magnetic force, which caused the electro
magnet to act like a dip-needle. The results obtained were on this
account very rough, but no evidence of any change in 6 could be
obtained even when an iron core was inserted in the coil, so as
to make it a powerful electromagnet.
If, therefore, a magnet contains matter in rapid rotation, the
ang'ular momentum of this rotation must be very small compared
with any quantities which we can measure, and we have as yet no
evidence of the existence of the terms Tme derived from their me
chanical action.
576.] Let us next consider the forces acting on the currents
of electricity, that is, the electromotive forces.
Let Y be the effective electromotive force due to induction, the
electromotive force which must act on the circuit from without
to balance it is Y'= — Yt and, by Lagrange's equation,
Y= -r= — — —.
dt dy dy
Since there are no terms in T involving the coordinate ^, the
second term is zero, and Y is reduced to its first term. Hence,
electromotive force cannot exist in a system at rest, and with con
stant currents.
Again, if we divide Y into three parts, Ym, Ye, and Yme, cor
responding to the three parts of T, we find that, since Tm does not
contain^, Ym = 0.
•W -C A V d dTe
We also find F, = — - , : -=-* •
dt dy
dT
Here -^-? is a linear function of the currents, and this part of
dy
the electromotive force is equal to the rate of change of this
function. This is the electromotive force of induction discovered
by Faraday. We shall consider it more at length afterwards.
577.] From the part of T, depending on velocities multiplied by
currents, we find Ymc = — ^- •
dt du
dT
Now — -j^ is a linear function of the velocities of the conductors.
dy
If, therefore, any terms of Tme have an actual existence, it would
be possible to produce an electromotive force independently of all
existing currents by simply altering the velocities of the conductors.
577-] ELECTROMOTIVE FORCE. 205
For instance, in the case of the suspended coil at Art. 559, if, when
the coil is at rest, we suddenly set it in rotation about the vertical
axis, an electromotive force would be called into action proportional
to the acceleration of this motion. It would vanish when the
motion became uniform, and be reversed when the motion was
retarded.
Now few scientific observations can be made with greater pre
cision than that which determines the existence or non-existence of
a current by means of a galvanometer. The delicacy of this method
far exceeds that of most of the arrangements for measuring the
mechanical force acting on a body. If, therefore, any currents could
be produced in this way they would be detected, even if they were
very feeble. They would be distinguished from ordinary currents
of induction by the following characteristics.
(1) They would depend entirely on the motions of the conductors,
and in no degree on the strength of currents or magnetic forces
already in the field.
(2) They would depend not on the absolute velocities of the con
ductors, but on their accelerations, and on squares and products of
velocities, and they would change sign when the acceleration be
comes a retardation, though the absolute velocity is the same.
Now in all the cases actually observed, the induced currents
depend altogether on the strength and the variation of currents in
the field, and cannot be excited in a field devoid of magnetic force
and of currents. In so far as they depend on the motion of con
ductors, they depend on the absolute velocity, and not on the change
of velocity of these motions.
We have thus three methods of detecting the existence of the
terms of the form Ttne, none of which have hitherto led to any
positive result. I have pointed them out with the greater care
because it appears to me important that we should attain the
greatest amount of certitude within our reach on a point bearing
so strongly on the true theory of electricity.
Since, however, no evidence has yet been obtained of such terms,
I shall now proceed on the assumption that they do not exist,
or at least that they produce no sensible effect, an assumption which
will considerably simplify our dynamical theory. We shall have
occasion, however, in discussing the relation of magnetism to light,
to shew that the motion which constitutes light may enter as a
factor into terms involving the motion which constitutes mag
netism.
CHAPTER VII.
THEORY OF ELECTRIC CIRCUITS.
578.] WE may now confine our attention to that part of the
kinetic energy of the system which depends on squares and products
of the strengths of the electric currents. We may call this the
Electrokinetic Energy of the system. The part depending on the
motion of the conductors belongs to ordinary dynamics, and we
have shewn that the part depending on products of velocities and
currents does not exist.
Let Al, AD &c. denote the different conducting circuits. Let
their form and relative position be expressed in terms of the variables
a?!, #2, &c., the number of which is equal to the number of degrees
of freedom of the mechanical system. We shall call these the
Geometrical Variables.
Let j/x denote the quantity of electricity which has crossed a given
section of the conductor A1 since the beginning of the time t. The
strength of the current will be denoted by y^, the fluxion of this
quantity.
We shall call y^ the actual current, and y^ the integral current.
There is one variable of this kind for each circuit in the system.
Let T denote the electrokinetic energy of the system. It is
a homogeneous function of the second degree with respect to the
strengths of the currents, and is of the form
T=±Llyl* + ±L2^+&c. + Ml2yly2 + &c.) (1)
where the coefficients L, M, &c. are functions of the geometrical
variables #15 #2, &c. The electrical variables yl} y2 do not enter
into the expression.
We may call Llt I/2, &c. the electric moments of inertia of the
circuits Alt A2, &c., and M12 the electric product of inertia of the
two circuits A^ and A2 , When we wish to avoid the language of
579-] ELECTROKINETIC MOMENTUM. 207
the dynamical theory, we shall call L^ the coefficient of self-induction
of the circuit Alt and M12 the coefficient of mutual induction of the
circuits A1 and A2. MlZ is also called the potential of the circuit
A^ with respect to Az. These quantities depend only on the form
and relative position of the circuits. We shall find that in the
electromagnetic system of measurement they are quantities of the
dimension of a line. See Art. 627.
By differentiating T with respect to y± we obtain the quantity _p1 ,
which, in the dynamical theory, may be called the momentum
corresponding to y±. In the electric theory we shall call p± the
electrokinetic momentum of the circuit A1 . Its value is
Pl = A ^1 + ^12^2 + &C"
The electrokinetic momentum of the circuit A1 is therefore made
up of the product of its own current into its coefficient of self-
induction, together with the sum of the products of the currents
in the other circuits, each into the coefficient of mutual induction
of A1 and that other circuit.
Electromotive Force.
579.] Let E be the impressed electromotive force in the circuit A,
arising from some cause, such as a voltaic or thermoelectric battery,
which would produce a current independently of magneto-electric
induction.
Let R be the resistance of the circuit, then, by Ohm's law, an
electromotive force Ey is required to overcome the resistance,
leaving an electromotive force E — Ry available for changing the
momentum of the circuit. Calling this force Y'9 we have, by the
general equations, dp dT
JL = -j- -- ^— >
at ay
but since T does not involve y, the last term disappears.
Hence, the equation of electromotive force is
or - =,+ •
The impressed electromotive force E is therefore the sum of two
parts. The first, JRy, is required to maintain the current y against
the resistance R. The second part is required to increase the elec
tromagnetic momentum p. This is the electromotive force which
must be supplied from sources independent of magneto-electric
208 LINEAR CIRCUITS. [580.
induction. The electromotive force arising from magneto -electric
induction alone is evidently — -j-, or, the rate of decrease of the
(A' v
electrokinetic momentum of the circuit.
Electromagnetic Force.
580.] Let X' be the impressed mechanical force arising from
external causes, and tending to increase the variable x. By the
general equations ^ d dT dT
dt dx dx
Since the expression for the electrokinetic energy does not contain
the velocity (#), the first term of the second member disappears,
and we find ^y
Ji. = -- 7 — •
dx
Here X' is the external force required to balance the forces arising
from electrical causes. It is usual to consider this force as the
reaction against the electromagnetic force, which we shall call X,
and which is equal and opposite to X'.
•v AT
Hence X = -T- >
dx
or, the electromagnetic force tending to increase any variable is equal
to the rate of increase of the electrokinetic energy per unit increase of
that variable, the currents being maintained constant.
If the currents are maintained constant by a battery during a
displacement in which a quantity, W, of work is done by electro
motive force, the electrokinetic energy of the system will be at the
same time increased by W. Hence the battery will be drawn upon
for a double quantity of energy, or 2 W, in addition to that which is
spent in generating heat in the circuit. This was first pointed out
by Sir W. Thomson*. Compare this result with the electrostatic
property in Art. 93.
Case of Two Circuits.
581.] Let AI be called the Primary Circuit, and A2 the Secondary
Circuit. The electrokinetic energy of the system may be written
where L and N are the coefficients of self-induction of the primary
* Nichol's Cyclopaedia of Physical Science, ed. 1860, Article, ' Magnetism, Dy
namical Relations of.'
582.] TWO CIRCUITS. 209
and secondary circuits respectively, and M is the coefficient of their
mutual induction.
Let us suppose that no electromotive force acts on the secondary
circuit except that due to the induction of the primary current.
We have then ci
E2 = B2fa+ (My, + Ny2] = 0.
Integrating this equation with respect to t, we have
Ry2 + Hjfi + Ny2 = C, a constant,
where y.^ is the integral current in the secondary circuit.
The method of measuring an integral current of short duration
will be described in Art. 748, and it is easy in most cases to ensure
that the duration of the secondary current shall be very short.
Let the values of the variable quantities in the equation at the
end of the time t be accented, then, if y^ is the integral current,
or the whole quantity of electricity which flows through a section
of the secondary circuit during the time t,
If the secondary current arises entirely from induction, its initial
value jr. 2 must be zero if the primary current is constant, and the
conductors at rest before the beginning of the time t.
If the time t is sufficient to allow the secondary current to die
away, y£y its final value, is also zero, so that the equation becomes
The integral current of the secondary circuit depends in this case
on the initial and final values
Induced Currents.
582.] Let us begin by supposing the primary circuit broken,
or y^ = 0, and let a current y{ be established in it when contact
is made.
The equation which determines the secondary integral current is
When the circuits are placed side by side, and in the same direc
tion, M is a positive quantity. Hence, when contact is made in
the primary circuit, a negative current is induced in the secondary
circuit.
When the contact is broken in the primary circuit, the primary
current ceases, and the induced current is y^ where
The secondary current is in this case positive.
VOL. II. P
210 LINEAR CIRCUITS.
If the primary current is maintained constant, and the form or
relative position of the circuits altered so that M becomes M', the
integral secondary current is y2, where
In the case of two circuits placed side by side and in the same
direction M diminishes as the distance between the circuits in
creases. Hence, the induced current is positive when this distance
is increased and negative when it is diminished.
These are the elementary cases of induced currents described in
Art. 530.
Mechanical Action between the Two Circuits.
583.] Let x be any one of the geometrical variables on which
the form and relative position of the circuits depend, the electro
magnetic force tending to increase x is
dL . dM . dN
If the motion of the system corresponding to the variation of x
is such that each circuit moves as a rigid body, L and N will be
independent of %, and the equation will be reduced to the form
dx
Hence, if the primary and secondary currents are of the same
sign, the force X, which acts between the circuits, will tend to
move them so as to increase M.
If the circuits are placed side by side, and the currents flow in
the same direction, M will be increased by their being brought
nearer together. Hence the force X is in this case an attraction.
584.] The whole of the phenomena of the mutual action of two
circuits, whother the induction of currents or the mechanical force
between them, depend on the quantity Jf, which we have called the
coefficient of mutual induction. The method of calculating this
quantity from the geometrical relations of the circuits is given in
Art. 524, but in the investigations of the next chapter we shall not
assume a knowledge of the mathematical form of this quantity.
We shall consider it as deduced from experiments on induction,
as, for instance, by observing the integral current when the
secondary circuit is suddenly moved from a given position to an
infinite distance, or to any position in which we know that M= 0.
CHAPTER VIII.
EXPLORATION OF THE FIELD BY MEANS OF THE SECONDARY
CIRCUIT.
585.] We have proved in Arts. 582, 583, 584 that the electro
magnetic action between the primary and the secondary circuit
depends on the quantity denoted by M, which is a function of the
form and relative position of the two circuits.
Although this quantity M is in fact the same as the potential
of the two circuits, the mathematical form and properties of which
we deduced in Arts. 423, 492, 521, 539 from magnetic and electro
magnetic phenomena, we shall here make no reference to these
results, but begin again from a new foundation, without any
assumptions except those of the dynamical theory as stated in
Chapter VII.
The electrokinetic momentum of the secondary circuit consists
of two parts (Art. 578), one, Milt depending on the primary current
ilt while the other, Niz, depends on the secondary current i2. We
are now to investigate the first of these parts, which we shall
denote by j?, where n _
We shall also suppose the primary circuit fixed, and the primary
current constant. The quantity jt?, the electrokinetic momentum of
the secondary circuit, will in this case depend only on the form
and position of the secondary circuit, so that if any closed curve
be taken for the secondary circuit, and if the direction along this
curve, which is to be reckoned positive, be chosen, the value of p
for this closed curve is determinate. If the opposite direction along
the curve had been chosen as the positive direction, the sign of
the quantity jo would have been reversed.
586.] Since the quantity p depends on the form and position
of the circuit, we may suppose that each portion of the circuit
212 ELECTROMAGNETIC FIELD.
contributes something1 to the value of p, and that the part con
tributed by each portion of the circuit depends on the form and
position of that portion only, and not on the position of other parts
of the circuit.
This assumption is legitimate, because we are not now considering
a current, the parts of which may, and indeed do, act on one an
other, but a mere circuit, that is, a closed curve along which a
current may flow, and this is a purely geometrical figure, the parts
of which cannot be conceived to have any physical action on each
other.
We may therefore assume that the part contributed by the
element ds of the circuit is Jds, where J is a quantity depending
on the position and direction of the element ds. Hence, the value
of p may be expressed as a line-integral
(2)
where the integration is to be extended once round the circuit.
587.] We have next to determine the form of the quantity «7~.
In the first place, if ds is reversed in direction, / is reversed in
sign. Hence, if two circuits ABCE and AECD
have the arc AEG common, but reckoned in
opposite directions in the two circuits, the sum
of the values of p for the two circuits
Fl*g- 35- and AECD will be equal to the value of p for
the circuit AJBCD, which is made up of the two circuits.
For the parts of the line-integral depending on the arc AEG are
equal but of opposite sign in the two partial circuits, so that they
destroy each other when the sum is taken, leaving only those parts of
the line- integral which depend on the external boundary of ABCD.
In the same way we may shew that if a surface bounded by a
closed curve be divided into any number of parts, and if the
boundary of each of these parts be considered as a circuit, the
positive direction round every circuit being the same as that round
the external closed curve, then the value of p for the closed curve is
equal to the sum of the values of p for all the circuits. See Art. 483.
588.] Let us now consider a portion of a surface, the dimensions
of which are so small with respect to the principal radii of curvature
of the surface that the variation of the direction of the normal
within this portion may be neglected. We shall also suppose that
if any very small circuit be carried parallel to itself from one part
of this surface to another, the value of p for the small circuit is
589.] ADDITION OF CIRCUITS. 213
not sensibly altered. This will evidently be the case if the dimen
sions of the portion of surface are small enough compared with
its distance from the primary circuit.
If any closed curve be drawn on this portion of the surface, the
value of p will be proportional to its area.
For the areas of any two circuits may be divided into small
elements all of the same dimensions, and having the same value
of p. The areas of the two circuits are as the numbers of these
elements which they contain, and the values of p for the two circuits
are also in the same proportion.
Hence, the value of p for the circuit which bounds any element
dS of a surface is of the form IdS,
where / is a quantity depending on the position of dS and on the
direction of its normal. We have therefore a new expression for p,
(3)
where the double integral is extended over any surface bounded by
the circuit.
589.] Let ABCD be a circuit, of which AC is an elementary
portion, so small that it may be considered straight.
Let APB and CQB be small equal areas in the
same plane, then the value of p will be the same
for the small circuits APB and CQB, or
p (APB) = p (CQB).
Hence p (APBQCD) = p (ABQCD) + p (APB),
= p (ABQCD) + 1
= p (ABCD), Fig. 36.
or the value of p is not altered by the substitution of the crooked
line APQCfor the straight line AC, provided the area of the circuit
is not sensibly altered. This, in fact, is the principle established
by Ampere's second experiment (Art. 506), in which a crooked
portion of a circuit is shewn to be equivalent to a straight portion
provided no part of the crooked portion is at a sensible distance
from the straight portion.
If therefore we substitute for the element ds three small elements,
dx, dy, and dz, drawn in succession, so as to form a continuous
path from the beginning to the end of the element ds, and if
Fdx, G dy, and II dz denote the elements of the line-integral cor
responding to dx, dy, and dz respectively, then
Jds = Fdse+ Gdy + Hdz. (4)
214 ELECTROMAGNETIC FIELD. [59°-
590.] We are now able to determine the mode in which the
quantity / dep3nds on the direction of the element ds. For,
by (4), f=P%. + 0*+H%. (5)
ds ds ds
This is the expression for the resolved part, in the direction of ds,
of a vector, the components of which, resolved in the directions of
the axes of x, y^ and z, are F, G, and H respectively.
If this vector be denoted by 51, and the vector from the origin
to a point of the circuit by p, the element of the circuit will be dp,
and the quaternion expression for / will be
We may now write equation (2) in the form
'
(7)
The vector 51 and its constituents F, G, H depend on the position
of ds in the field, and not on the direction in which it is drawn.
They are therefore functions of x, y, z, the coordinates of ds, and
not of I, m} n, its direction-cosines.
The vector 51 represents in direction and magnitude the time-
integral of the electromotive force which a particle placed at the
point (x, y, z) would experience if the primary current were sud
denly stopped. We shall therefore call it the Electrokinetic Mo
mentum at the point (x, ?/, z}. It is identical with the quantity
which we investigated in Art. 405 under the name of the vector-
potential of magnetic induction.
The electrokinetic momentum of any finite line or circuit is the
line-integral, extended along the line or circuit, of the resolved
part of the electrokinetic momentum at each point of the same.
591.] Let us next determine the value of
p for the elementary rectangle ABCD, of
which the sides are dy and dz, the positive
direction being from the direction of the
axis of y to that of z.
Let the coordinates of 0, the centre of
gravity of the element, be a?0, yQ, ZQ, and let
-p. 37 GQ> HQ be the values of G and of H at this
point.
The coordinates of A, the middle point of the first side of the
MAGNETIC INDUCTION. 215
rectangle, are yQ and ZQ — - dz. The corresponding value of G is
(8)
and the part of the value of p which arises from the side A is
approximately i dG
1 rlTT
Similarly, for B, H0dz+--^- Ay dz.
For (7, -G,dy-\d^dydz.
For D, — H0 dz + - — Ay dz.
2 cly
Adding these four quantities, we find the value of p for the
rectangle m da
If we now assume three new quantities, #, b, c, such that
dH dG i
>
(A)
a — -= -- -=-9
d dz
dF dH
-j --- j-
dz dx
dG dF
7 ~~ 7 '
dx dy J
and consider these as the constituents of a new vector 33, then, by
Theorem IV, Art. 24, we may express the line-integral of 51 round
any circuit in the form of the surface-integral of 33 over a surface
bounded by the circuit, thus
p = F~-^G +H~ds=(la + mb + nc}dS, (11)
J ^ ds ds ds' JJ
or p = JT 2t cose ds = f j T<& cos TJ d8, (12)
where e is the angle between 5( and ds, and rj that between 33 and
the normal to dS, whose direction-cosines are I, m, n, and T 51, T 33
denote the numerical values of 51 and 33.
Comparing this result with equation (3), it is evident that the
quantity / in that equation is equal to 33 cos r;, or the resolved part
of 33 normal to dS.
592.] We have already seen (Arts. 490, 541) that, according to
Faraday's theory, the phenomena of electromagnetic force and
216 ELECTROMAGNETIC FIELD. [593-
induction in a circuit depend on the variation of the number of
lines of magnetic induction which pass through the circuit. Now
the number of these lines is expressed mathematically by the
surface-integral of the magnetic induction through any surface
bounded by the circuit. Hence, we must regard the vector 23
and its components a, b, c as representing what we are already
acquainted with as the magnetic induction and its components.
In the present investigation we propose to deduce the properties
of this vector from the dynamical principles stated in the last
chapter, with as few appeals to experiment as possible.
In identifying this vector, which has appeared as the result of
a mathematical investigation, with the magnetic induction, the
properties of which we learned from experiments on magnets, we
do not depart from this method, for we introduce no new fact into
the theory, we only give a name to a mathematical quantity, and
the propriety of so doing is to be judged by the agreement of the
relations of the mathematical quantity with those of the physical
quantity indicated by the name.
The vector 33, since it occurs in a surface-integral, belongs
evidently to the category of fluxes described in Art. 13. The
vector 51, on the other hand, belongs to the category of forces,
since it appears in a line-integral.
593.] We must here recall to mind the conventions about positive
and negative quantities and directions, some of which were stated
in Art. 23. We adopt the right-handed system of axes, so that if
a right-handed screw is placed in the direction of the axis of x,
and a nut on this screw is turned in the positive direction of
rotation, that is, from the direction of y to that of z, it will move
along the screw in the positive direction of x.
We also consider vitreous electricity and austral magnetism as
positive. The positive direction of an electric current, or of a line
of electric induction, is the direction in which positive electricity
moves or tends to move, and the positive direction of a line of
magnetic induction is the direction in which a compass needle
points with the end which turns to the north. See Fig. 24, Art.
498, and Fig. 25, Art. 501.
The student is recommended to select whatever method appears
to him most effectual in order to fix these conventions securely in
his memory, for it is far more difficult to remember a rule which
determines in which of two previously indifferent ways a statement
is to be made, than a rule which selects one way out of many.
594-] THEORY OF A SLIDING PIECE. 217
594.] We have next to deduce from dynamical principles the
expressions for the electromagnetic force acting on a conductor
carrying an electric current through the magnetic field, and for
the electromotive force acting on the electricity within a body
moving in the magnetic field. The mathematical method which
we shall adopt may be compared with the experimental method
used by Faraday * in exploring the field by means of a wire, and
with what we have already done at Art. 490, by a method founded
on experiments. What we have now to do is to determine the
effect on the value of ji, the electroldnetic momentum of the
secondary circuit, due to given alterations of the form of that
circuit.
Let AA', BB' be two parallel straight conductors connected by
the conducting arc (7, which may be of any form, and by a straight
Fig. 38.
conductor AB, which is capable of sliding parallel to itself along
the conducting rails AA and BB'.
Let the circuit thus formed be considered as the secondary cir
cuit, and let the direction ABC be assumed as the positive direction
round it.
Let the sliding piece move parallel to itself from the position AB
to the position AB'. We have to determine the variation of _p, the
electrokinetic momentum of the circuit, due to this displacement
of the sliding piece.
The secondary circuit is changed from ABC to A'IfC, hence, by
Art. 587, p (AB'C)-p (ABC) = p (AA'B'B). (13)
We have therefore to determine the value of p for the parallel
ogram AA'BB. If this parallelogram is so small that we may
neglect the variations of the direction and magnitude of the mag
netic induction at different points of its plane, the value of p is,
by Art. 591, 33 cos r\ . AA'ffBj where 33 is the magnetic induction,
* Exp. Res., 3082, 3087, 3113.
218 ELECTROMAGNETIC FIELD. [595-
and 77 the angle which it makes with the positive direction of the
normal to the parallelogram AA'B'B.
We may represent the result geometrically by the volume of the
parallelepiped, whose base is the parallelogram AA'B'B, and one of
whose edges is the line AM, which represents in direction and
magnitude the magnetic induction 33. If the parallelogram is in
the plane of the paper, and if AM is drawn upwards from the paper,
the volume of the parallelepiped is to be taken positively, or more
generally, if the directions of the circuit AB, of the magnetic in
duction AM, and of the displacement AA', form a right-handed
system when taken in this cyclical order.
The volume of this parallelepiped represents the increment of
the value of p for the secondary circuit due to the displacement
of the sliding piece from AB to A'B'.
Electromotive Force acting on the Sliding Piece.
595.] The electromotive force produced in the secondary circuit
by the motion of the sliding piece is, by Art. 579,
If we suppose AA' to be the displacement in unit of time, then
AA' will represent the velocity, and the parallelepiped will represent
~, and therefore, by equation (14), the electromotive force in the
Ctu
negative direction B A.
Hence, the electromotive force acting on the sliding piece AB,
in consequence of its motion through the magnetic field, is repre
sented by the volume of the parallelepiped, whose edges represent
in direction and magnitude — the velocity, the magnetic induction,
and the sliding piece itself, and is positive when these three direc
tions are in right-handed cyclical order.
Electromagnetic Force acting on the Sliding Piece.
596.] Let i2 denote the current in the secondary circuit in the
positive direction ABC, then the work done by the electromagnetic
force on AB while it slides from the position AB to the position
A'B' is (M'—M)ili2, where M and M' are the values of M12 in
the initial and final positions of AB. But (M'—M)^ is equal
to//— p, and this is represented by the volume of the parallelepiped
on AB, AM, and AA'. Hence, it' we draw a line parallel to AB
598.] LINES OF MAGNETIC INDUCTION. 219
to represent the quantity AB.i2, the parallelepiped contained by
this line, by AM, the magnetic induction, and by A A, the displace
ment, will represent the work done during- this displacement.
For a given distance of displacement this will be greatest when
the displacement is perpendicular to the parallelogram whose sides
are AB and AM. The electromagnetic force is therefore represented
by the area of the parallelogram on AB and AM multiplied by ?/2,
and is in the direction of the normal to this parallelogram, drawn so
that AB, AM, and the normal are in right-handed cyclical order.
Four Definitions of a Line of Magnetic Induction.
597.] If the direction AA ', in which the motion of the sliding
piece takes place, coincides with AM, the direction of the magnetic
induction, the motion of the sliding piece will not call electromotive
force into action, whatever be the direction of AB, and if AB carries
an electric current there will be no tendency to slide along AA.
Again,, if AB} the sliding piece, coincides in direction with AM,
the direction of magnetic induction, there will be no electromotive
force called into action by any motion of AB, and a current through
AB will not cause AB to be acted on by mechanical force.
We may therefore define a line of magnetic induction in four
different ways. It is a line such that —
(1) If a conductor be moved along it parallel to itself it will
experience no electromotive force.
(2) If a conductor carrying a current be free to move along a
line of magnetic induction it will experience no tendency to do so.
(3) If a linear conductor coincide in direction with a line of
magnetic induction, and be moved parallel to itself in any direction,
it will experience no electromotive force in the direction of its
length.
(4) If a linear conductor carrying an electric current coincide
in direction with a line of magnetic induction it will not experience
any mechanical force.
General Equations of Electromotive Force.
598.] We have seen that E, the electromotive force due to in
duction acting on the secondary circuit, is equal to j- , where
220 ELECTROMAGNETIC FIELD.
To determine the value of E, let us differentiate the quantity
under the integral sign with respect to ^, remembering that if the
secondary circuit is in motion, as, y, and z are functions of the time.
We obtain
f(dF tfa dG_dy dH dz.
J^dt ds + dt r& + <fc «fr'
C,dF dx dG dy dH dz^ dx
J ^ dx ds dx ds dx ds ' dt
dF dx dG dy dHdz^ dy
dy ds dy ds dy ds' dt
dF dx_ dG_dy dH dz. dz
ds dz ds dz ds' dt
-/<
ds dt ds dt
,2,
Now consider the second term of the integral, and substitute
from equations (A), Art. 591, the values of — and -7- . This term
dx dx
then becomes,
[( ^ 7)^z dF dx dF dy dF dz^dx
J\Cdi" ds "f 'das ds + ^7 Ts + Hz ds' di •
which we may write
f f (ty 7 dz dF^ dx _
— / (C '/ — ^ 7- + T-J -T7 ^-
J ^ ds ds ds ' dt
Treating the third and fourth terms in the same way, and col-
i ,. .-, . dx dy - dz
lectmg the terms m - , ^ , and — , remembering that
dx
^
= F~7- , (3)
dt ~ dsdt>" L dt
and therefore that the integral, when taken round the closed
curve, vanishes,
f ( dz dx dG. dy
/ (a^7 ~c^7 7-^ 7
J ^ dt dt dt ) ds
dx d dH dz
598.] ELECTROMOTIVE FORCE. 221
We may write this expression in the form
Equations of
Electromotive (-B)
Force.
dy -.dz dF d^
where P = c -~ — o-j = =-
dz dx dG d^J
dt dt dt dy
_ dx dy dH d^
The terms involving the new quantity ^ are introduced for the
sake of giving generality to the expressions for P, Q, R. They
disappear from the integral when extended round the closed circuit.
The quantity ^ is therefore indeterminate as far as regards the
problem now before us, in which the total electromotive force round
the circuit is to be determined. We shall find, however, that when
we know all the circumstances of the problem, we can assign a
definite value to ^, and that it represents, according to a certain
definition, the electric potential at the point x, y, z.
The quantity under the integral sign in equation (5) represents
the electromotive force acting on the element ds of the circuit.
If we denote by T @, the numerical value of the resultant of P,
Q, and R, and by e, the angle between the direction of this re
sultant and that of the element ds, we may write equation (5),
JT<$ cost els. (6)
fi =JT<$ cost els.
The vector @ is the electromotive force at the moving element
ds. Its direction and magnitude depend on the position and
motion of ds, and on the variation of the magnetic field, but not
on the direction of ds. Hence we may now disregard the circum
stance that ds forms part of a circuit, and consider it simply as a
portion of a moving body, acted on by the electromotive force Q.
The electromotive force at a point has already been defined in
Art. 68. It is also called the resultant electrical force, being the
force which would be experienced by a unit of positive electricity
placed at that point. We have now obtained the most general
value of this quantity in the case of a body moving in a magnetic
field due to a variable electric system.
If the body is a conductor, the electromotive force will produce a
current ; if it is a dielectric, the electromotive force will produce
only electric displacement.
222 ELECTROMAGNETIC FIELD. [599-
The electromotive force at a point, or on a particle, must be
carefully distinguished from the electromotive force along an arc
of a curve, the latter quantity being the line-integral of the former.
See Art, 69.
599.] The electromotive force, the components of which are
defined by equations (B), depends on three circumstances. The first
of these is the motion of the particle through the magnetic field.
The part of the force depending on this motion is expressed by the
first two terms on the right of each equation. It depends on the
velocity of the particle transverse to the lines of magnetic induction.
If © is a vector representing the velocity, and 33 another repre
senting the magnetic induction, then if (^ is the part of the elec
tromotive force depending on the motion,
^ = V. ® 33, (7)
or, the electromotive force is the vector part of the product of the
magnetic induction multiplied by the velocity, that is to say, the
magnitude of the electromotive force is represented by the area
of the parallelogram, whose sides represent the velocity and the
magnetic induction, and its direction is the normal to this parallel
ogram, drawn so that the velocity, the magnetic induction, and the
electromotive force are in right-handed cyclical order.
The third term in each of the equations (B) depends on the time-
variation of the magnetic field. This may be due either to the
time-variation of the electric current in the primary circuit, or to
motion of the primary circuit. Let (£2 be the part of the electro
motive force which depends on these terms. Its components are
dF dG dH
-w ~w and -w
and these are the components of the vector, — — or 21. Hence,
dt
6, = -& (8)
The last term of each equation (B) is due to the variation of the
function ^ in different parts of the field. We may write the third
part of the electromotive force, which is due to this cause,
@3 = - V*. (9)
The electromotive force, as defined by equations (B), may therefore
be written in the quaternion form,
@= r.® 33-21- V*. (10)
600.] MOVING AXES. 223
On the Modification of the Equations of Electromotive Force when the
Axes to which they are referred are moving in Space.
600.] Let #', y', / be the coordinates of a point referred to a
system of rectangular axes moving- in space, and let #, ?/, z be the
coordinates of the same point referred to fixed axes.
Let the components of the velocity of the origin of the moving
system be u, v, w, and those of its angular velocity w^ o>2, co3
referred to the fixed system of axes, and let us choose the fixed
axes so as to coincide at the given instant with the moving ones,
then the only quantities which will be different for the two systems
of axes will be those differentiated with respect to the time. If
bx
•— denotes a component velocity of a point moving in rigid con-
o t
nexion with the moving axes, and - - and -j- that of any moving
ci/t civ
point, having the same instantaneous position, referred to the fixed
and the moving axes respectively, then
dx __ §x duo , ^
~di = bi + ~di'
with similar equations for the other components.
By the theory of the motion of a body of invariable form,
bx
— = « + wa*
}> (2)
Since F is a component of a directed quantity parallel to x,
if — r— be the value of -=- referred to the moving axes,
dl" (ZFbv dFby clFbz dF
Substituting for -=- and -y- their values as deduced from the
dy dz
equations (A) of magnetic induction, and remembering that, by (2),
d bx d ly d bz
= °' = a>3' =~^
_b_x d^b^ d_by dffbz d bz
dt ~ dx U dx bt + dx U fy bt + ~dx~ ~U + dx *i
b , bz dF
224 ELECTKOMAGNETIC FIELD. [6OI.
Ifweaowput
dF' dV *z dF
_^=j H
Of 01 Of
-.
The equation for P, the component of the electromotive force
parallel to a?, is, by (B),
referred to the fixed axes. Substituting the values of the quanti
ties as referred to the moving axes, we have
dy> dz> dF d(* + V) (9)
Cdt~^Tt"~dt~ dx
for the value of P referred to the moving axes.
601.] It appears from this that the electromotive force is ex
pressed by a formula of the same type, whether the motions of the
conductors be referred to fixed axes or to axes moving in space, the
only difference between the formulae being that in the case of
moving axes the electric potential # must be changed into vI/ + 4//.
In all cases in which a current is produced in a conducting cir
cuit, the electromotive force is the line-integral
taken round the curve. The value of * disappears from this
integral, so that the introduction of SP' has no influence on its
value. In all phenomena, therefore, relating to closed circuits and
the currents in them, it is indifferent whether the axes to which we
refer the system be at rest or in motion. See Art. 668.
On the Electromagnetic Force acting on a Conductor which carries
an Electric Current through a Magnetic Field.
602.] We have seen in the general investigation, Art. 583, that if
a?x is one of the variables which determine the position and form of
the secondary circuit, and if XL is the force acting on the secondary
circuit tending to increase this variable, then
. ,-v
Since ^ is independent of xlf we may write
602.] ELECTROMAGNETIC FORCE. 225
(3)
and we have for the value of Xlf
ds
Now let us suppose that the displacement consists in moving
every point of the circuit through a distance b% in the direction
of #, b% being any continuous function of s, so that the different
parts of the circuit move independently of each other, while the
circuit remains continuous and closed.
Also let X be the total force in the direction of x acting on
the part of the circuit from s = 0 to s = s, then the part corre-
7 ~V
spending to the element ds will be -=- ds. We shall then have the
following expression for the work done by the force during the
displacement,
/dX ^ f d / ~.dx ~dy -rTdz\ „
rbatk s= LI -j— ( F-7~ + G -f + J2V) 6# ds, (4)
ds 2J dbsn^ ds ds ds'
where the integration is to be extended round the closed curve,
remembering that 80? is an arbitrary function of s. We may there
fore perform the differentiation with respect to b x in the same
way that we differentiated with respect to t in Art. 598, remem
bering that dx dy dz
-= - = 1, -y£- = 0. and -= — = 0. (5)
dbx '
We thus find
The last term vanishes when the integration is extended round
the closed curve, and since the equation must hold for all forms
of the function bas, we must have
dX . / dy -, dz\ /P,N
— = »| ((?_£_£._), (7)
ds 2V ds ds'
an equation which gives the force parallel to x on any element of
the circuit. The forces parallel to y and z are
dT . f dz dx\ .
— = lAa— -- C-=-)* (8)
d* 2V ds ds'
dZ . ^dx dy^ , .
•j- = 4f^-3 — «•/•!• (9)
ds 2\ ds dx'
The resultant force on the element is given in direction and mag
nitude by the quaternion expression i2Vdp$$, where i2 is the
numerical measure of the current, and dp and 53 are vectors
VOL. II. Q
226 ELECTROMAGNETIC FIELD. [603.
representing the element of the circuit and the magnetic in
duction, and the multiplication is to be understood in the Hamil-
tonian sense.
603.] If the conductor is to be treated not as a line but as a
body, we must express the force on the element of length, and the
current through the complete section, in terms of symbols denoting
the force per unit of volume, and the current per unit of area.
Let X, Y, Z now represent the components of the force referred to
unit of volume, and u, v, w those of the current referred to unit of
area. Then, if S represents the section of the conductor, which we
shall suppose small, the volume of the element ds will be Sds, and
n = -^ - - . Hence, equation (7) will become
S(vc-w6), (10)
(Equations of
Electromagnetic (C)
or X = vc —wb.
Similarly Y= wa — uc,
, r/ 7 Force.
and Z — ub — va.
Here X, J", Z are the components of the electromagnetic force on
an element of a conductor divided by the volume of that element ;
n, v, w are the components of the electric current through the
element referred to unit of area, and #, b, c are the components
of the magnetic induction at the element, which are also referred
to unit of area.
If the vector § represents in magnitude and direction the force
acting on unit of volume of the conductor, and if (£ represents the
electric current flowing through it,
en)
CHAPTER IX.
GENERAL EQUATIONS OF THE ELECTROMAGNETIC FIELD.
604.] IN our theoretical discussion of electrodynamics we began
by assuming- that a system of circuits carrying electric currents
is a dynamical system, in which the currents may be regarded as
velocities, and in which the coordinates corresponding to these
velocities do not themselves appear in the equations. It follows
from this that the kinetic energy of the system, so far as it depends
on the currents, is a homogeneous quadratic function of the currents,
in which the coefficients depend only on the form and relative
position of the circuits. Assuming these coefficients to be known,
by experiment or otherwise, we deduced, by purely dynamical rea
soning, the laws of the induction of currents, and of electromagnetic
attraction. In this investigation we introduced the conceptions
of the electrokinetic energy of a system of currents, of the electro
magnetic momentum of a circuit, and of the mutual potential of
two circuits.
We then proceeded to explore the field by means of various con
figurations of the secondary circuit, and were thus led to the
conception of a vector 2[, having a determinate magnitude and
direction at any given point of the field. We called this vector the
electromagnetic momentum at that point. This quantity may be
considered as the time-integral of the electromotive force which
would be produced at that point by the sudden removal of all the
currents from the field. It is identical with the quantity already
investigated in Art. 405 as the vector-potential of magnetic in
duction. Its components parallel to x, y, and z are F, G, and H.
The electromagnetic momentum of a circuit is the line-integral
of $1 round the circuit.
We then, by means of Theorem IV, Art. 24, transformed the
Q 2
228 GENERAL EQUATIONS. [605.
line-integral of £1 into the surface-integral of another vector, 53,
whose components are a, d, c, and we found that the phenomena
of induction due to motion of a conductor, and those of electro
magnetic force can be expressed in terms of 53. We gave to 53
the name of the Magnetic induction, since its properties are iden
tical with those of the lines of magnetic induction as investigated
by Faraday.
We also established three sets of equations : the first set, (A),
are those of magnetic induction, expressing it in terms of the elec
tromagnetic momentum. The second set, (B), are those of electro
motive force, expressing it in terms of the motion of the conductor
across the lines of magnetic induction, and of the rate of variation
of the electromagnetic momentum. The third set, (C), are the
equations of electromagnetic force,, expressing it in terms of the
current and the magnetic induction.
The current in all these cases is to be understood as the actual
current, which includes not only the current of conduction, but the
current due to variation of the electric displacement.
The magnetic induction 53 is the quantity which we have already
considered in Art. 400. In an unmagnetized body it is identical
with the force on a unit magnetic pole, but if the body is mag
netized, either permanently or by induction, it is the force which
would be exerted on a unit pole, if placed in a narrow crevasse in
the body, the walls of which are perpendicular to the direction of
magnetization. The components of 53 are #, #, c.
It follows from the equations (A), by which a, b, c are defined,
that da M (i^^
dx dy dz
This was shewn at Art. 403 to be a property of the magnetic
induction.
605.] We have defined the magnetic force within a magnet, as
distinguished from the magnetic induction, to be the force on a
unit pole placed in a narrow crevasse cut parallel to the direction of
magnetization. This quantity is denoted by ȣ), and its components
by a, /3, y. See Art. 398.
If 3 is the intensity of magnetization, and A, B, C its com
ponents, then, by Art. 400,
a = a -f 4 TT A,
c = y+4-n C.
(Equations of Magnetization.) (D)
6o6.] MAGNETIC EQUATIONS. 229
We may call these the equations of magnetization, and they
indicate that in the electromagnetic system the magnetic induction
33, considered as a vector, is the sum, in the Hamiltonian sense, of
two vectors, the magnetic force .£), and the magnetization 3 multi
plied by 47T, or 33 = £ + 4?r3.
In certain substances, the magnetization depends on the magnetic
force, and this is expressed by the system of equations of induced
magnetism given at Arts. 426 and 435.
606.] Up to this point- of our investigation we have deduced
everything from purely dynamical considerations, without any
reference to quantitative experiments in electricity or magnetism.
The only use we have made of experimental knowledge is to re
cognise, in the abstract quantities deduced from the theory, the
concrete quantities discovered by experiment, and to denote them
by names which indicate their physical relations rather than their
mathematical generation.
In this way we have pointed out the existence of the electro
magnetic momentum §1 as a vector whose direction and magnitude
vary from one part of space to another, and from this we have
deduced, by a mathematical process, the magnetic induction, 33, as
a derived vector. We have not, however, obtained any data for
determining either 51 or 33 from the distribution of currents in the
field. For this purpose we must find the mathematical connexion
between these quantities and the currents.
We begin by admitting the existence of permanent magnets,
the mutual action of which satisfies the principle of the conservation
of energy. We make no assumption with respect to the laws of
magnetic force except that which follows from this principle,
namely, that the force acting on a magnetic pole must be capable
of being derived from a potential.
We then observe the action between currents and magnets, and
we find that a current acts on a magnet in a manner apparently the
same as another magnet would act if its strength, form, and position
were properly adjusted, and that the magnet acts on the current
in the same way as another current. These observations need not
be supposed to be accompanied with actual measurements of the
forces. They are not therefore to be considered as furnishing
numerical data, but are useful only in suggesting questions for
our consideration.
The question these observations suggest is, whether the magnetic
field produced by electric currents, as it is similar to that produced
230 GENERAL EQUATIONS. [607.
by permanent magnets in many respects, resembles it also in being-
related to a potential ?
The evidence that an electric circuit produces, in the space sur
rounding it, magnetic effects precisely the same as those produced
by a magnetic shell bounded by the circuit, has been stated in
Arts. 482-485.
We know that in the case of the magnetic shell there is a
potential, which has a determinate value for all points outside the
substance of the shell, but that the values of the potential at two
neighbouring points, on opposite sides of the shell,, differ by a finite
quantity.
If the magnetic field in the neighbourhood of an electric current
resembles that in the neighbourhood of a magnetic shell, the
magnetic potential, as found by a line-integration of the magnetic
force, will be the same for any two lines of integration, provided
one of these lines can be transformed into the other by continuous
motion without cutting the electric current.
If, however, one line of integration cannot be transformed into
the other without cutting the current, the line-integral of the
magnetic force along the one line will differ from that along the
other by a quantity depending on the strength of the current. The
magnetic potential due to an electric current is therefore a function
having an infinite series of values with a common difference, the
particular value depending on the course of the line of integration.
Within the substance of the conductor, there is no such thing as
a magnetic potential.
607.] Assuming that the magnetic action of a current has a
magnetic potential of this kind, we proceed to express this result
mathematically.
In the first place, the line-integral of the magnetic force round
any closed curve is zero, provided the closed curve does not surround
the electric current.
In the next place, if the current passes once, and only once,
through the closed curve in the positive direction, the line-integral
has a determinate value, which may be used as a measure of the
strength of the current. For if the closed curve alters its form
in any continuous mariner without cutting the current, the line-
integral will remain the same.
In electromagnetic measure, the line-integral of the magnetic
force round a closed curve is numerically equal to the current
through the closed curve multiplied by 4 TT.
607.] ELECTRIC CURRENTS. 231
If we take for the closed curve the parallelogram whose sides
are dy and dz, the line-integral of the magnetic force round the
parallelogram is ^y dp
^dy dz
and if u, vf w are the components of the flow of electricity, the
current through the parallelogram is
u dy dz.
Multiplying this by 47r, and equating the result to the line-
integral, we obtain the equation
dy dz
with the similar equations
do, dy ( (Equations of /-™\
4 7T V = -= ~- ) Electric Currents.) W
dz dx
dp da
dx dy J
which determine the magnitude and direction of the electric currents
when the magnetic force at every point is given.
When there is no current, these equations are equivalent to the
condition that adx + fi dy + y dz = D£l,
or that the magnetic force is derivable from a magnetic potential
in all points of the field where there are no currents.
By differentiating the equations (E) with respect to x, y, and z
respectively, and adding the results, we obtain the equation
du dv dw
. I I . Q
dx dy dz
which indicates that the current whose components are u, v, w is
subject to the condition of motion of an incompressible fluid, and
that it must necessarily flow in closed circuits.
This equation is true only if we take #, v, and w as the com
ponents of that electric flow which is due to the variation of electric
displacement as well as to true conduction.
We have very little experimental evidence relating to the direct
electromagnetic action of currents due to the variation of electric
displacement in dielectrics, but the extreme difficulty of reconciling
the laws of electromagnet ism with the existence of electric currents
which are not closed is one reason among many why we must admit
the existence of transient currents due to the variation of displace
ment. Their importance will be seen when we come to the electro
magnetic theory of light.
232 GENERAL EQUATIONS. [6o8.
608.] We have now determined the relations of the principal
quantities concerned in the phenomena discovered by Orsted, Am
pere, and Faraday. To connect these with the phenomena described
in the former parts of this treatise, some additional relations are
necessary.
When electromotive force acts on a material body, it produces
in it two electrical effects, called by Faraday Induction and Con
duction, the first being most conspicuous in dielectrics, and the
second in conductors.
In this treatise, static electric induction is measured by what we
have called the electric displacement, a directed quantity or vector
which we have denoted by £), and its components by/*, #, k.
In isotropic substances, the displacement is in the same direction
as the electromotive force which produces it, and is proportional
to it, at least for small values of this force. This may be expressed
by the equation i
<T\ -IT- rr, (Equation of Electric /-pry
4 IT ' Displacement.)
where ^is the dielectric capacity of the substance. See Art. 69.
In substances which are not isotropic, the components /, #, h of
the electric displacement 2) are linear functions of the components
P, Q, -K of the electromotive force (£.
The form of the equations of electric displacement is similar to
that of the equations of conduction as given in Art. 298.
These relations may be expressed by saying that K is, in isotropic
bodies, a scalar quantity, but in other bodies it is a linear and vector
function, operating on the vector (£.
609.] The other effect of electromotive force is conduction. The
laws of conduction as the result of electromotive force were esta
blished by Ohm, and are explained in the second part of this
treatise, Art. 241. They may be summed up in the equation
ft = C (£, (Equation of Conductivity.) (G)
where (£ is the intensity of the electromotive force at the point,
$ is the density of the current of conduction, the components of
which are p, q, r, and C is the conductivity of the substance, which,
in the case of isotropic substances, is a simple scalar quantity, but
in other substances becomes a linear and vector function operating
on the vector ($. The form of this function is given in Cartesian
coordinates in Art. 298.
610.] One of the chief peculiarities of this treatise is the doctrine
which it asserts, that the true electric current (£, that on which the
614.] CURRENTS OF DISPLACEMENT. 233
electromagnetic phenomena depend, is not the same thing as $, the
current of conduction, but that the time- variation of 2), the electric
displacement, must be taken into account in estimating the total
movement of electricity, so that we must write,
(£ = £+2), (Equation of True Currents.) (H)
or, in terms of the components,
dt
dg
j V
dk
(H*)
611.] Since both $ and 2) depend on the electromotive force ($,
we may express the true current (£ in terms of the electromotive
force, thus
or, in the case in which C and K are constants,
w = CR+ —- KC-j-'
47T dt
612.] The volume-density of the free electricity at any point
is found from the components of electric displacement by the
equation ^f dg dk
613.] The surface-density of electricity is
where /, m, n are the direction-cosines of the normal drawn from
the surface into the medium in which f, g, li are the components of
the displacement, and /', m' ', n' are those of the normal drawn from
the surface into the medium in which they are f', /, //.
614.] When the magnetization of the medium is entirely induced
by the magnetic force acting on it, we may write the equation of
induced magnetization, $$ = /*«£), (L)
where p is the coefficient of magnetic permeability, which may
be considered a scalar quantity, or a linear and vector function
operating on «£j, according as the medium is isotropic or not.
234
GENEKAL EQUATIONS.
615.] These may be regarded as the principal relations among
the quantities we have been considering. They may be combined
so as to eliminate some of these quantities, but our object at present
is not to obtain compactness in the mathematical formulae, but
to express every relation of which we have any knowledge. To
eliminate a quantity which expresses a useful idea would be rather
a loss than a gain in this stage of our enquiry.
There is one result, however, which we may obtain by combining
equations (A) and (E), and which is of very great importance.
If we suppose that no magnets exist in the field except in the
form of electric circuits, the distinction which we have hitherto
maintained between the magnetic force and the magnetic induction
vanishes, because it is only in magnetized matter that these quan
tities differ from each other.
According to Ampere's hypothesis, which will be explained in
Art. 833, the properties of what we call magnetized matter are due
to molecular electric circuits, so that it is only when we regard the
substance in large masses that our theory of magnetization is
applicable, and if our mathematical methods are supposed capable
of taking account of what goes on within the individual molecules,
they will discover nothing but electric circuits, and we shall find
the magnetic force and the magnetic induction everywhere identical.
In order, however, to be able to make use of the electrostatic or of
the electromagnetic system of measurement at pleasure we shall
retain the coefficient //, remembering that its value is unity in the
electromagnetic system.
616.] The components of the magnetic induction are by equa
tions (A), Art. 591, dH dG
n — —
a/ — — -y-
dy dz
dF dH
o — — --- —
dz dx
dF
dx dy
The components of the electric current are by equations (E),
Art. 607,
dy aft
4 77 U — V- 7- >
0* &
da
-
dz
d(B
~
dx
dy
=£
dx
da
~~
dy
6l6.]
VECTOR-POTENTIAL OP CURRENTS.
According to our hypothesis a, b, c are identical with
respectively. We therefore obtain
If we write
235
i, fift /uy
tffo? dy dy2 dz2
dF dG dH
J = -j- + -r + ~r >
ax dy dz
dzdx
we may write equation (1),
Similarly,
dJ
4 TT ja v = -- + V2 #»
If we write F'=- fff U- dx dy dz, ~|
-, j
where r is the distance of the given point from the element xy z,
and the integrations are to be extended over all space, then
(7)
The quantity x. disappears from the equations (A), and it is not
related to any physical phenomenon. If we suppose it to be zero
everywhere, / will also be zero everywhere, and equations (5),
omitting the accents, will give the true values of the components
of 51.
* The negative sign is employed here in order to make our expressions consistent
with those in which Quaternions are employed.
236 GENERAL EQUATIONS. [617.
617.] We may therefore adopt, as a definition of 2[, that it
is the vector-potential of the electric current, standing1 in the same
relation to the electric current that the scalar potential stands to
the matter of which it is the potential, and obtained by a similar
process of integration, which may be thus described. —
From a given point let a vector be drawn, representing1 in mag
nitude and direction a given element of an electric current, divided
by the numerical value of the distance of the element from the
given point. Let this be done for every element of the electric
current. The resultant of all the vectors thus found is the poten
tial of the whole current. Since the current is a vector quantity,
its potential is also a vector. See Art. 422.
When the distribution of electric currents is given, there is one,
and only one, distribution of the values of 31, such that 31 is every
where finite and continuous, and satisfies the equations
V2§1= 47Tf*<£, fl.VSl = 0,
and vanishes at an infinite distance from the electric system. This
value is that given by equations (5), which may be written
Quaternion Expressions for tJie Electromagnetic Equations.
618.] In this treatise we have endeavoured to avoid any process
demanding from the reader a knowledge of the Calculus of Qua
ternions. At the same time we have not scrupled to introduce the
idea of a vector when it was necessary to do so. When we have
had occasion to denote a vector by a symbol, we have used a
German letter, the number of different vectors being so great that
Hamilton's favourite symbols would have been exhausted at once.
Whenever therefore, a German letter is used it denotes a Hamil-
tonian vector, and indicates not only its magnitude but its direction.
The constituents of a vector are denoted by Roman or Greek letters.
The principal vectors which we have to consider are : —
Constituents.
The radius vector of a point .................. p x y z
The electromagnetic momentum at a point 2[ F G H
The magnetic induction ..................... 53 a I c
The (total) electric current .................. (£ u v w
The electric displacement ..................... 2) f g h
6 1 9.] QUATEKNION EXPRESSIONS. 237
Constituents.
The electromotive force ..................... (£ P Q R
The mechanical force ........................ g XYZ
The velocity of a point ........................ © or p so y z
The magnetic force ........................... «£) a /3 y
The intensity of magnetization ............ 3 ABC
The current of conduction .................. ft p q r
We have also the following scalar functions : —
,The electric potential ^.
The magnetic potential (where it exists) 12.
The electric density e.
The density of magnetic ' matter ' m.
Besides these we have the following quantities, indicating physical
properties of the medium at each point : —
(7, the conductivity for electric currents.
K, the dielectric inductive capacity.
fji, the magnetic inductive capacity.
These quantities are, in isotropic media, mere scalar functions
of p, but in general they are linear and vector operators on the
vector functions to which they are applied. K and JJL are certainly
always self- conjugate, and C is probably so also.
619.] The equations (A) of magnetic induction, of which the
first is> dH dG
a = -= --- r-»
dy dz
may now be written sg _ yyty
where V is the operator
. d . d -, d
%-j- +7-7- + £-7-1
dx * dy dz
and Vindicates that the vector part of the result of this operation
is to be taken.
Since 21 is subject to the condition $ V 2[ = 0, V§[ is a pure
vector, and the symbol V is unnecessary.
The equations (B) of electromotive force, of which the first is
, . dF d*
P = cy—oz -- - --- r- ,
dt dx
become @= F®33 — $ — V*.
The equations (C) of mechanical force, of which the first is
v , d^> dil
JL = cv — mv — e — -- m -7— j
dx dx
become = 7 $ 33 —
238 GENERAL EQUATIONS. [619.
The equations (D) of magnetization, of which the first is
a — a 4- 4 TT A,
become 33 — <$ 4- 4 TT 3.
The equations (E) of electric currents, of which the first is
dy d(3
4 TT u — -/ -- fi
dy dz
become 4 -n & =
The equation of the current of conduction is, by Ohm's Law,
£ = <7<g.
That of electric displacement is
3) = -?-K®.
4 7T
The equation of the total current, arising from the variation of
the electric displacement as well as from conduction, is
<£ - S + 2X
When the magnetization arises from magnetic induction,
SB = M£.
We have also, to determine the electric volume-density,
e = £V$).
To determine the magnetic volume-density,
•m = S V 3.
When the magnetic force can be derived from a potential
= - V 12.
CHAPTER X.
DIMENSIONS OF ELECTRIC UNITS.
620.] EVERY electromagnetic quantity may be defined with
reference to the fundamental units of Length, Mass, and Time.
If we begin with the definition of the unit of electricity, as given
in Art. 65, we may obtain definitions of the units of every other
electromagnetic quantity, in virtue of the equations into which
they enter along with quantities of electricity. The system of
units thus obtained is called the Electrostatic System.
If, on the other hand, we begin with the definition of the unit
magnetic pole, as given in Art. 374, we obtain a different system
of units of the same set of quantities. This system of units is
not consistent with the former system, and is called the Electro
magnetic System.
We shall begin by stating those relations between the different
units which are common to both systems, and we shall then form
a table of the dimensions of the units according to each system.
621.] We shall arrange the primary quantities which we have
to consider in pairs. In the first three pairs, the product of the
two quantities in each pair is a quantity of energy or work. In
the second three pairs, the product of each pair is a quantity of
energy referred to unit of volume.
FIRST THREE PAIRS.
Electrostatic Pair.
Symbol.
( 1 ) Quantity of electricity . . . . e
(2) Line-integral of electromotive force, or electric po
tential E
240 DIMENSIONS OF UNITS. [622.
Magnetic Pair.
Symbol.
(3) Quantity of free magnetism, or strength of a pole . m
(4) Magnetic potential ...... H
ElectroJcinetic Pair.
(5) Electroldnetic momentum of a circuit . . p
(6) Electric current ....... C
SECOND THREE PAIRS.
Electrostatic Pair.
(7) Electric displacement (measured by surface-density) . 3)
(8) Electromotive force at a point . . . (£'
Magnetic Pair.
(9) Magnetic induction * ..... 33
(10) Magnetic force .;•« » ..... $
Electrokinetic Pair.
(11) Intensity of electric current at a point . . . (£
(12) Vector potential of electric currents . . .51
622.] The following relations exist between these quantities.
In the first place, since the dimensions of energy are , and
those of energy referred to unit of volume , we have the
following equations of dimensions :
(1)
(2)
Secondly, since e, p and 51 are the time-integrals of C, fi, and (£
Thirdly, since E, 12, and p are the line-integrals of @, .£>, and 91
respectively,
Finally, since et C, and m are the surface-integrals of $), 6, and
respectively,
625.] THE TWO SYSTEMS OF UNITS. 241
623.] These fifteen equations are not independent, and in order
to deduce the dimensions of the twelve units involved, we require
one additional equation. If, however, we take either e or m as an
independent unit, we can deduce the dimensions of the rest in
terms of either of these.
(3) and (5) [j,] = M=
(4) and (6)
(10)
624.] The relations of the first ten of these quantities may be
exhibited by means of the following arrangement : —
e, 2), «£), C and 12. E (£, 33, m and p.
The quantities in the first line are derived from e by the same
operations as the corresponding quantities in the second line are
derived from m. It will be seen that the order of the quantities
in the first line is exactly the reverse of the order in the second
line. The first four of each line have the first symbol in the
numerator. The second four in each line have it in the deno
minator.
All the relations given above are true whatever system of units
we adopt.
625.] The only systems of any scientific value are the electro
static and the electromagnetic system. The electrostatic system is
VOL. II. ft
242 DIMENSIONS OF UNITS. [626.
founded on the definition of the unit of electricity, Arts. 41, 42,
and may be deduced from the equation
which expresses that the resultant force (£ at any point, due to the
action of a quantity of electricity e at a distance L, is found by
dividing e by 7/2. Substituting the equations of dimension (1) and
(8), we find
whence \e\ = \L* If* T^} , m =
in the electrostatic system.
The electromagnetic system is founded on a precisely similar
definition of the unit of strength of a magnetic pole, Art. 374,
leading to the equation ^ m
* : = L* '
J/
whence
e-] ri
^J - \-^ J
and [e] =
in the electromagnetic system. From these results we find the
dimensions of the other quantities.
626.] Table of Dimensions.
Dimensions in
c, , , Electrostatic Electromagnetic
Symbol Sygtem System
Quantity of electricity .... e [Z* M * T~l] \L* M*\.
Line-integral of electro- | ^ ^ M- T~^ \ti H* T~*\.
motive force 3
Quantity of magnetism -\
Electrokinetic momentum t . $m I [tf M*\ \L* M* T~1].
of a circuit ) *
Electric current C [L* M* T
Magnetic potential ) ' {Q,
Electric displacement | _ [T-^M^T~l~[ IT'
Surface-density
Electromotive force at a point @ [^"M/^7-1] [Z*Jtf* I7"2].
Magnetic induction 53 [IT^*] [i;-*^^-1].
Magnetic force § [L* M* T~*] [L~* M* I'1].
Strength of current at a point (£ [Z~* If * T"2] [^~^ If* T~l] .
Vector potential 31 [Z-*!f*]
628.] TABLE OF DIMENSIONS. 243
627.] We have already considered the products of the pairs of
these quantities in the order in which they stand. Their ratios are
in certain cases of scientific importance. Thus
Electrostatic Electromagnetic
Symbol. System. System.
e l~T2~\
-=- = capacity of an accumulator . . q [Z] T~ I
/•coefficient of self-induction *\
-^- = j of a circuit, or electro- > L \~T~\ \f\*
(. magnetic capacity J
2) _ ( specific inductive capacity | ^ r _
¥=: ( of dielectric \
33 r^72!
-£- = magnetic inductive capacity . . ju y2 M-
4P L^ J
x? r- yr —i p T — 1
-— = resistance of a conductor .... R -=- "TT
(S C specific resistance of a )
"T = : | substance }'
628.] If the units of length, mass, and time are the same in the
two systems, the number of electrostatic units of electricity con
tained in one electromagnetic unit is numerically equal to a certain
velocity, the absolute value of which does not depend on the
magnitude of the fundamental units employed. This velocity is
an important physical quantity, which we shall denote by the
symbol v.
Number of Electrostatic Units in one Electromagnetic Unit.
For*, C, 11, 5), £, (£, v.
Form, ^ .0, 93, <£, 21, -•
v
For electrostatic capacity, dielectric inductive capacity, and con
ductivity, v*.
For electromagnetic capacity, magnetic inductive capacity, and
resistance, —5- •
p2
Several methods of determining the velocity v will be given in
Arts. 768-780.
In the electrostatic system the specific dielectric inductive capa
city of air is assumed equal to unity. This quantity is therefore
represented by -^ in the electromagnetic system.
R 2,
244 DIMENSIONS OF UNITS. [629.
In the electromagnetic system the specific magnetic inductive
capacity of air is assumed equal to unity . This quantity is there
fore represented by —$• in the electrostatic system.
Practical System of Electric Units.
629.] Of the two systems of units, the electromagnetic is of the
greater use to those practical electricians who are occupied with
electromagnetic telegraphs. If, however, the units of length, time,
and mass are those commonly used in other scientific work, such
as the metre or the centimetre, the second, and the gramme, the
units of resistance and of electromotive force will be so small that
to express the quantities occurring in practice enormous numbers
must be used, and the units of quantity and capacity will be so
large that only exceedingly small fractions of them can ever occur
in practice. Practical electricians have therefore adopted a set of
electrical units deduced by the electromagnetic system from a large
unit of length and a small unit of mass.
The unit of length used for this purpose is ten million of metres,
or approximately the length of a quadrant of a meridian of the
earth.
The unit of time is, as before, one second.
The unit of mass is 10~~n gramme, or one hundred millionth
part of a milligramme.
The electrical units derived from these fundamental units have
been named after eminent electrical discoverers. Thus the practical
unit of resistance is called the Ohm, and is represented by the
resistance-coil issued by the British Association, and described in
Art. 340. It is expressed in the electromagnetic system by a
velocity of 10,000,000 metres per second.
The practical unit of electromotive force is called the Volt, and
is not very different from that of a DanielPs cell. Mr. Latimer
Clark has recently invented a very constant cell, whose electro
motive force is almost exactly 1.457 Volts.
The practical unit of capacity is called the Farad. The quantity
of electricity which flows through one Ohm under the electromotive
force of one Volt during one second, is equal to the charge produced
in a condenser whose capacity is one Farad by an electromotive
force of one Volt.
The use of these names is found to be more convenient in practice
than the constant repetition of the words ' electromagnetic units,'
629.]
PEACTICAL UNITS.
245
with the additional statement of the particular fundamental units
on which they are founded.
When very large quantities are to be measured, a large unit
is formed by multiplying the original unit by one million, and
placing before its name the prefix mega.
In like manner by prefixing micro a small unit is formed, one
millionth of the original unit.
The following table gives the values of these practical units in
the different systems which have been at various times adopted.
FUNDAMENTAL
UNITS.
PRACTICAL
SYSTEM.
B. A. REPORT,
1863.
THOMSON.
WEBER.
Length,
Time,
Mass.
Earth's Quadrant,
Second,
10-11 Gramme.
Metre,
Second,
Gramme.
Centimetre,
Second,
Gramme.
Millimetre,
Second,
Milligramme.
Resistance
Ohm
IO7
IO9
IO1
Electromotive force
Volt
IO5
IO8
10U
Capacity
Quantity
Farad
Farad
(charged to a Volt.)
io-7
io-2
io-9
io-1
io-10
10
CHAPTER XL
ON ENERGY AND STRESS IN THE ELECTROMAGNETIC FIELD.
Electrostatic Energy.
630.] THE energy of the system may be divided into the Potential
Energy and the Kinetic Energy.
The potential energy due to electrification has been already con
sidered in Art. 85. It may be written
r=is(**), (i)
where e is the charge of electricity at a place where the electric
potential is ty, and the summation is to be extended to every place
where there is electrification.
If fj ffj Ji are the components of the electric displacement, the
quantity of electricity in the element of volume dx dy dz is
where the integration is to be extended throughout all space.
631.] Integrating this expression by parts, and remembering
that when the distance, r, from a given point of a finite electrified
system becomes infinite, the potential ty becomes an infinitely small
quantity of the order r*1, and that/, g, h become infinitely small
quantities of the order r~2, the expression is reduced to
where the integration is to be extended throughout all space.
If we now write P, Q, R for the components of the electromotive
dty d^ city
force, instead of -- — , -- — and -- =- , we find
dx dy dz
(5)
633-] MAGNETIC ENERGY. 247
Hence, the electrostatic energy of the whole field will be the same
if we suppose that it resides in every part of the field where elec
trical force and electrical displacement occur, instead of being
confined to the places where free electricity is found.
The energy in unit of volume is half the product of the electro
motive force and the electric displacement, multiplied by the cosine
of the angle which these vectors include.
In Quaternion language it is —4/9(5 3).
Magnetic Energy.
632.] We may treat the energy due to magnetization in a similar
way. If A, J5, C are the components of magnetization and a, /3, y
the components of magnetic force, the potential energy of the
system of magnets is, by Art. 389,
Cy]dxdydzt (6)
the integration being extended over the space occupied by mag
netized matter. This part of the energy, however, will be included
in the kinetic energy in the form in which we shall presently
obtain it.
633.] We may transform this expression when there are no elec
tric currents by the following method.
We know that da db do
Hence, by Art. 97, if
cm d& cm
f. o .. ( R\
as is always the case in magnetic phenomena where there are no
currents,
' =0, (9)
the integral being extended throughout all space, or
jjl{(a + lTtA)a + (P + lTtB)p + (y+±'nC)y}dxdydz = 0. (10)
Hence, the energy due to a magnetic system
248 ENERGY AND STRESS. [634.
Electrokinetic Energy.
634.] We have already, in Art. 578, expressed the kinetic energy
of a system of currents in the form
T=\^(pi\ (12).
where p is the electromagnetic momentum of a circuit, and % is
the strength of the current flowing round it, and the summation
extends to all the circuits.
But we have proved, in Art. 590, that p may be expressed as
a line-integral of the form
where F, G, H are the components of the electromagnetic mo-
mentum, §C, at the point (xy z), and the integration is to be ex
tended round the closed circuit s. We therefore find
2 *"' J \ £?<$ ds ds'
If ^, z;, w are the components of the density of the current at
any point of the conducting circuit, and if S is the transverse
section of the circuit, then we may write
. dx .dy . dz
i — = uS, i^ = vS, 2-v = ^£, (15)
ds ds ds
and we may also write the volume
Sds = dxdydz,
and we now find _
T = i / // (Fu + Gv + Hw) dxdydz, (16)
where the integration is to be extended to every part of space
where there are electric currents.
635.] Let us now substitute for u, v, w their values as given by
the equations of electric currents (E), Art. 607, in terms of the
components a, /3, y of the magnetic force. We then have
where the integration is extended over a portion of space including
all the currents.
If we integrate this by parts, and remember that, at a great
distance r from the system, a, /3, and y are of the order of mag
nitude r~3, we find that when the integration is extended through
out all space, the expression is reduced to
/^7 dH\ fflG dF\] 7
637.] ELECTROKINETIC ENERGY. 249
By the equations (A), Art. 591, of magnetic induction, we may
substitute for the quantities in small brackets the components of
magnetic induction a, b, c, so that the kinetic energy may be
written 1 /././.
T= — JJJ(aa + 6p + cy)da!dydz9 (19)
where the integration is to be extended throughout every part of
space in which the magnetic force and magnetic induction have
values differing from zero.
The quantity within brackets in this expression is the product of
the magnetic induction into the resolved part of the magnetic force
in its own direction.
In the language of quaternions this may be written more simply,
where 33 is the magnetic induction, whose components are «, b, c,
and JQ is the magnetic force, whose components are a, (3, y.
636.] The electrokinetic energy of the system may therefore be
expressed either as an integral to be taken where there are electric
currents, or as an integral to be taken over every part of the field
in which magnetic force exists. The first integral, however, is the
natural expression of the theory which supposes the currents to act
upon each other directly at a distance, while the second is appro
priate to the theory which endeavours to explain the action between
the currents by means of some intermediate action in the space
between them. As in this treatise we have adopted the latter
method of investigation, we naturally adopt the second expression
as giving the most significant form to the kinetic energy.
According to our hypothesis, we assume the kinetic energy to
exist wherever there is magnetic force, that is, in general, in every
part of the field. The amount of this energy per unit of volume
is -- '— S S3 $3, and this energy exists in the form of some kind
o 77
of motion of the matter in every portion of space.
When we come to consider Faraday's discovery of the effect of
magnetism on polarized light, we shall point out reasons for be
lieving that wherever there are lines of magnetic force, there is
a rotatory motion of matter round those lines. See Art. 821.
Magnetic and Electrokinetic Energy compared.
637.] We found in Art. 423 that the mutual potential energy
250 ENERGY AND STRESS. [638.
of two magnetic shells, of strengths $ and $', and bounded by the
closed curves s and / respectively, is
cos e ,
— as as ,
where e is the angle between the directions of ds and ds', and r
is the distance between them.
We also found in Art. 521 that the mutual energy of two circuits
s and /, in which currents i and i' flow, is
-if
cos e 7 .. f
ds ds .
If i, i' are equal to (/>, </>' respectively, the mechanical action
between the magnetic shells is equal to that between the cor
responding electric circuits, and in the same direction. In the case
of the magnetic shells, the force tends to diminish their mutual
potential energy, in the case of the circuits it tends to increase their
mutual energy, because this energy is kinetic.
It is impossible, by any arrangement of magnetized matter, to
produce a system corresponding in all respects to an electric circuit,
for the potential of the magnetic system is single valued at every
point of space, whereas that of the electric system is many- valued.
But it is always possible, by a proper arrangement of infinitely
small electric circuits, to produce a system corresponding in all
respects to any magnetic system, provided the line of integration
which we follow in calculating the potential is prevented from
passing through any of these small circuits. This will be more
fully explained in Art. 833.
The action of magnets at a distance is perfectly identical with
that of electric currents. We therefore endeavour to trace both
to the same cause, and since we cannot explain electric currents
by means of magnets, we must adopt the other alternative, and
explain magnets by means of molecular electric currents.
638.J In our investigation of magnetic phenomena, in Part III
of this treatise, we made no attempt to account for magnetic action
at a distance, but treated this action as a fundamental fact of
experience. We therefore assumed that the energy of a magnetic
system is potential energy, and that this energy is diminished when
the parts of the system yield to the magnetic forces which act
on them.
If, however, we regard magnets as deriving their properties from
electric currents circulating within their molecules, their energy
639-] AMPERE'S THEORY OF MAGNETS. 251
is kinetic, and the force between them is such that it tends to
move them in a direction such that if the strengths of the currents
were maintained constant the kinetic energy would increase.
This mode of explaining magnetism requires us also to abandon
the method followed in Part III, in which we regarded the magnet
as a continuous and homogeneous body, the minutest part of which
has magnetic properties of the same kind as the whole.
We must now regard a magnet as containing a finite, though
very great, number of electric circuits, so that it has essentially
a molecular, as distinguished from a continuous structure.
If we suppose our mathematical machinery to be so coarse that
our line of integration cannot thread a molecular circuit, and that
an immense number of magnetic molecules are contained in our
element of volume, we shall still arrive at results similar to those
of Part III, but if we suppose our machinery of a finer order,
and capable of investigating all that goes on in the interior of the
molecules, we must give up the old theory of magnetism, and adopt
that of Ampere, which admits of no magnets except those which
consist of electric currents.
We must also regard both magnetic and electromagnetic energy
as kinetic energy, and we must attribute to it the proper sign,
as given in Art. 635.
In what follows, though we may occasionally, as in Art. 639, &c.,
attempt to carry out the old theory of magnetism, we shall find
that we obtain a perfectly consistent system only when we abandon
that theory and adopt Ampere^s theory of molecular currents, as in
Art. 644.
The energy of the field therefore consists of two parts only, the
electrostatic or potential energy
W = \jjj(Pf +
and the electromagnetic or kinetic energy
T= ~
ON THE FORCES WHICH ACT ON AN ELEMENT OF A BODY PLACED
IN THE ELECTROMAGNETIC FIELD.
Forces acting on a Magnetic Element.
639.] The potential energy of the element dx dy dz of a body
magnetized with an intensity whose components are A, B, C, and
252 ENERGY AND STRESS. [640.
placed in a field of magnetic force whose components are a, /3, y, is
Hence, if the force urging the element to move without rotation
in the direction of a? is X1dxdydz,
and if the moment of the couple tending to turn the element about
the axis of x from y towards z is L dxdydz,
L = By-C($. (2)
The forces and the moments corresponding to the axes of y and
z may be written down by making the proper substitutions.
640. J If the magnetized body carries an electric current, of
which the components are u3 v, w, then, by equations C, Art. 60S,
there will be an additional electromagnetic force whose components
are X2, Y%, ZZ) of which X2 is
X2 = VG — wb. (3)
Hence, the total force, X, arising from the magnetism of the
molecule, as well as the current passing through it, is
+vc-«6. (4)
dx dx
The quantities a, 6, c are the components of magnetic induction,
and are related to a, (3, y, the components of magnetic force, by
the equations given in Art. 400,
a = a -f 4 TT A,
£=/3 + 477.£, (5)
C = 7+477(7.
The components of the current, u, v, w, can be expressed in terms
of a, /3, y by the equations of Art. 607,
dy d(3
4 TT u — —- j-
dy dz
da dy
4;TTV = -= -~-
dz dx
dp da
TT 4/7rw = -f- — -T
Hence dx dy
(6)
_
' dx } dx n dx
1 ( da -.da da 1 d 1
= — \a T +b—+c~---- (a*+(32 +y2)}- (7)
47T ( dx dy dz 2 dee. }
641.] THEORY OF STRESS. 253
Multiplying this equation, (8), by a, and dividing by 47i, we may
add the result to (7), and we find
(9)
also, by (2), i = ((J-/3) y-(c-y)/3), (10)
= ~(iv-eft), (11)
where X is the force referred to unit of volume in the direction of
#, and L is the moment of the forces about this axis.
On the Explanation of these Forces by the Hypothesis of a Medium
in a State of Stress.
641 .] Let us denote a stress of any kind referred to unit of area
by a symbol of the form Phk) where the first suffix, h, indicates that
the normal to the surface on which the stress is supposed to act
is parallel to the axis of h, and the second suffix, ft , indicates that
the direction of the stress with which the part of the body on
the positive side of the surface acts on the part on the negative
side is parallel to the axis of k.
The directions of h and k may be the same, in which case the
stress is a normal stress. They may be oblique to each other, in
which case the stress is an oblique stress, or they may be perpen
dicular to each other, in which case the stress is a tangential
stress.
The condition that the stresses shall not produce any tendency
to rotation in the elementary portions of the body is
P - P
^hk — rWi'
In the case of a magnetized body, however, there is such a
tendency to rotation, and therefore this condition, which holds in
the ordinary theory of stress, is not fulfilled.
Let us consider the effect of the stresses on the six sides of
the elementary portion of the body dx dy dz, taking the origin of
coordinates at its centre of gravity.
On the positive face dy dz, for which the value of % is \ dx, the
forces are —
254
ENERGY AND STRESS.
[641.
Parallel to x,
dP.
Parallel to y, (Pxy + * -^f dx} dydz = Y+x, .
(12)
Parallel to
(P«+ 4
The forces acting on the opposite side, — X_X9 —Y_x) and — Z_x,
may be found from these by changing the sign of dx. We may
express in the same way the systems of three forces acting on each
of the other faces of the element, the direction of the force being
indicated by the capital letter, and the face on which it acts by
the suffix.
If Xdxdydz is the whole force parallel to x acting on the element,
Xdxdydz = XH
,£P.
whence
d
dx dx
^P + ^
dy vx dz
(13)
If Ldxdydz is the moment of the forces about the axis of x
tending to turn the element from y to 0,
Ldxdydz =
whence L = Pyg — Pzy . (14)
Comparing the values of X and L given by equations (9) and
(11) with those given by (13) and (14), we find that, if we make
= --_(aa-±(<S
1
TTJ
1
p —
-*-%* — A ~
~k
= ~T-C^
+r
i
4 77
1
= ^va"/'
I
(15)
the force arising from a system of stress of which these are the
components will be statically equivalent, in its effects on each
642.]
MAGNETIC STRESS.
255
element of the body, with the forces arising from the magnetization
and electric currents.
642.] The nature of the stress of which these are the components
may be easily found, by making the axis of x bisect the angle
between the directions of the magnetic force and the magnetic
induction, and taking the axis of y in the plane of these directions,
and measured towards the side of the magnetic force.
If we put <£) for the numerical value of the magnetic force, 33 for
that of the magnetic induction, and 2 € for the angle between their
directions,
a = *y cos e, /3 = «£) sin e, y •=. 0,
a — 33 cos e, b = — 33 sin e, c
1 - 2 i '2
4 jf
(17)
p _ p — p _ p _ o
yz ~~ zx zy — -*• xz
Pxv = — - 33 <£> cos e sin e,
Pyx = — — - 33 4p cos e sin e.
Hence, the state of stress may be considered as compounded of —
(1) A pressure equal in all directions = -— «&2.
8 77
(2) A tension along the line bisecting the angle between the
directions of the magnetic force and the magnetic induction
-
(3) A pressure along the line bisecting the exterior angle between
these directions = — 33 § sin2 e.
(4) A couple tending to turn every element of the substance in
the plane of the two directions from the direction of magnetic
induction to the direction of magnetic force — - - 33 <£) sin 2 e.
When the magnetic induction is in the same direction as the
magnetic force, as it always is in fluids and non-magnetized solids,
then e = 0, and making the axis of x coincide with the direction of
the magnetic force,
256
ENERGY AND STRESS.
[643.
(18)
and the tangential stresses disappear.
The stress in this case is therefore a hydrostatic pressure - - «£j2,
combined with a longitudinal tension — 33 <£) along the lines of
f 4 TT
force.
643.] When there is no magnetization, 33 = $3, and the stress is
still further simplified, being a tension along the lines of force equal
to -— <£)2, combined with a pressure in all directions at right angles
. 1
to the lines of force, numerically equal also to - — 43 2- The com
ponents of stress in this important case are
Pxx = —(a*-(3*-y
P = — ( 2-a2-/3
** 8 77 ^
yz zy ^^
(19)
PX = Px = JLal3t
4 7T
The force arising from these stresses on an element of the medium
referred to unit of volume is
d d
f -J-PVZ+ -rP™>
ay " dz
Y_ d
=
1 C da d/3 dyl 1 ( d(3 dal ' 1 C dy da)
^da d(3 dy\ 1 /da dy\
__
dy
fa
dy
Now
da d(3 dy
-7- + ~r + -T
dx dy dz
da. dy
-j- -y-
dz dx
dft da
-j =- = 4 77 W-
ax dy
where m is the density of austral magnetic matter referred to unit
645-] TENSION ALONG LINES OF FORCE. 257
of volume, and v and w are the components of electric currents
referred to unit of area perpendicular to y and z respectively. Hence,
X = am+ vy — wj3
Similarly Y = fim + wa— uy,
(Equations of
Electromagnetic (20)
Force.)
Zi = ym-i-vip — va.
644.] If we adopt the theories of Ampere and Weber as to the
nature of magnetic and diamagnetic bodies, and assume that mag
netic and diamagnetic polarity are due to molecular electric currents,
we get rid of imaginary magnetic matter, and find that everywhere
* = 0,and *? + *0 + ?y=0, (21)
dx dy dz
so that the equations of electromagnetic force become,
X = v y — w /3,
Y—wa-uy} (22)
Z = ujB—va.
These are the components of the mechanical force referred to unit
of volume of the substance. The components of the magnetic force
are a, /3, y, and those of the electric current are u, v, w. These
equations are identical with those already established. (Equations
(C), Art, 603.)
645.] In explaining the electromagnetic force by means of a
state of stress in a medium, we are only following out the con
ception of Faraday"*, that the lines of magnetic force tend to
shorten themselves, and that they repel each other when placed
side by side. All that we have done is to express the value of
the tension along the lines, and the pressure at right angles to
them, in mathematical language, and to prove that the state of
stress thus assumed to exist in the medium will actually produce
the observed forces on the conductors which carry electric currents.
We have asserted nothing as yet with respect to the mode
in which this state of stress is originated and maintained in the
medium. We have merely shewn that it is possible to conceive
the mutual action of electric currents to depend on a particular
kind of stress in the surrounding medium, instead of being a direct
and immediate action at a distance.
Any further explanation of the state of stress, by means of the
motion of the medium or otherwise, must be regarded as a separate
and independent part of the theory, which may stand or fall without
affecting our present position. See Art. 832.
* Esrp. Res., 3266, 3267, 3268.
VOL. TT. S
258 ENERGY AND STRESS. [646.
In the first part of this treatise, Art. 108, we shewed that the
observed electrostatic forces may be conceived as operating through
the intervention of a state of stress in the surrounding medium.
We have now done the same for the electromagnetic forces, and
it remains to be seen whether the conception of a medium capable
of supporting these states of stress is consistent with other known
phenomena, or whether we shall have to put it aside as unfruitful.
In a field in which electrostatic as well as electromagnetic action
is taking place, we must suppose the electrostatic stress described
in Part I to be superposed on the electromagnetic stress which we
have been considering.
646.] If we suppose the total terrestrial magnetic force to be
10 British units (grain, foot, second), as it is nearly in Britain, then
the tension perpendicular to the lines of force is 0.128 grains weight
per square foot. The greatest magnetic tension produced by Joule *
by means of electromagnets was about 140 pounds weight on the
square inch.
* Sturgeon's Annals of Electricity, vol. v. p. 187 (1840) ; or Philosophical Magazine,
Dec., 1851.
CHAPTER XII.
CURRENT-SHEETS.
647.] A CURRENT-SHEET is an infinitely thin stratum of con
ducting matter, bounded on both sides by insulating1 media, so that
electric currents may flow in the sheet, but cannot escape from it
except at certain points called Electrodes, where currents are made
to enter or to leave the sheet.
In order to conduct a finite electric current, a real sheet must
have a finite thickness, and ought therefore to be considered a
conductor of three dimensions. In many cases, however, it is
practically convenient to deduce the electric properties of a real
conducting sheet, or of a thin layer of coiled wire, from those of
a current-sheet as defined above.
We may therefore regard a surface of any form as a current-sheet.
Having selected one side of this surface as the positive side, we
shall always suppose any lines drawn on the surface to be looked
at from the positive side of the surface. In the case of a closed
surface we shall consider the outside as positive. See Art. 294,
where, however, the direction of the current is defined as seen from
the negative side of the sheet.
The Current -function.
648.] Let a fixed point A on the surface be chosen as origin, and
let a line be drawn on the surface from A to another point P. Let
the quantity of electricity which in unit of time crosses this line
from left to right be $, then </> is called the Current-function at
the point P.
The current-function depends only on the position of the point P,
and is the same for any two forms of the line AP, provided this
s z
260 CURRENT-SHEETS. [649.
line can be transformed by continuous motion from one form to the
other without passing through an electrode. For the two forms of
the line will enclose an area within which there is no electrode, and
therefore the same quantity of electricity which enters the area across
one of the lines must issue across the other.
If s denote the length of the line AP, the current across ds from
left to right will be — ds.
If </> is constant for any curve, there is no current across it. Such
a curve is called a Current-line or a Stream-line.
649.] Let \}f be the electric potential at any point of the sheet,
then the electromotive force along any element ds of a curve will be
d^ ,
f-d*,
ds
provided no electromotive force exists except that which arises from
differences of potential.
If \^ is constant for any curve, the curve is called an Equi-
potential Line.
650.] We may now suppose that the position of a point on the
sheet is defined by the values of </> and \[r at that point. Let dsl be
the length of the element of the equipotential line ^ intercepted
between the two current lines <£ and <j> + d<l>, and let ds2 be the
length of the element of the current line $ intercepted between the
two equipotential lines ty and \fr + d\lf. We may consider ds} and dsz
as the sides of the element dty d\^r of the sheet. The electromotive
force — d\l/ in the direction of ds2 produces the current d<p across dslf
Let the resistance of a portion of the sheet whose length is ds2t
and whose breadth is dsl} be ds2
(T —.- J
0*1
where <r is the specific resistance of the sheet referred to unit of
area, then ds.2 7
'*-*zf'*
, ds-, ds.2
whence jj- = <r yf -
a<j) d\l/
651.] If the sheet is of a substance which conducts equally well
in all directions, dsl is perpendicular to ds2. In the case of a sheet
of uniform resistance or is constant, and if we make \jr' = a\f/, we
shall have ds: __ d(j>
d9t~~ d+'*
and the stream-lines and equipotential lines will cut the surface into
little squares.
652.] MAGNETIC POTENTIAL. 261
It follows from this that if fa and i/r/ are conjugate functions
(Art. 183) of cj) and \f/t the curves fa may be stream-lines in the
sheet for which the curves x/// are the corresponding equipotential
lines. One case, of course, is that in which fa = \f/' and \j/i = — <£.
In this case the equipotential lines become current-lines, and the
current-lines equipotential lines *.
If we have obtained the solution of the distribution of electric
currents in a uniform sheet of any form for any particular case, we
may deduce the distribution in any other case by a proper trans
formation of the conjugate functions, according to the method given
in Art. 190.
652.] We have next to determine the magnetic action of a
current-sheet in which the current is entirely confined to the sheet,
there being no electrodes to convey the current to or from the
sheet.
In this case the current-function 0 has a determinate value at
every point, and the stream-lines are closed curves which do not
intersect each other, though any one stream-line may intersect
itself.
Consider the annular portion of the sheet between the stream
lines $ and <j)-{-b<p. This part of the sheet is a conducting circuit
in which a current of strength 8 $ circulates in the positive direction
round that part of the sheet for which c/> is greater than the given
value. The magnetic effect of this circuit is the same as that of
a magnetic shell of strength 8 $ at any point not included in the
substance of the shell. Let us suppose that the shell coincides with
that part of the current-sheet for which 0 has a greater value than
it has at the given stream-line.
By drawing all the successive stream-lines, beginning with that
for which $ has the greatest value, and ending with that for which
its value is least, we shall divide the current-sheet into a series
of circuits. Substituting for each circuit its corresponding mag
netic shell, we find that the magnetic effect of the current-sheet
at any point not included in the thickness of the sheet is the same
as that of a complex magnetic shell, whose strength at any point
is C-{-(f), where C is a constant.
If the current-sheet is bounded, then we must make C 4- <£ = 0
at the bounding curve. If the sheet forms a closed or an infinite
surface, there is nothing to determine the value of the constant C.
* See Thomson, Camb. and Dub. Math. Journ., vol. iii. p. 286.
262 CURRENT -SHEETS. [653.
653.] The magnetic potential at any point on either side of the
current-sheet is given, as in Art. 415, by the expression
= ^-
where r is the distance of the given point from the element of
surface dS, and Q is the angle between the direction of r, and that
of the normal drawn from the positive side of dS.
This expression gives the magnetic potential for all points not
included in the thickness of the current-sheet, and we know that
for points within a conductor carrying a current there is no such
thing as a magnetic potential.
The value of H is discontinuous at the current-sheet, for if &j_
is its value at a point just within the current-sheet, and Q,2 its
value at a point close to the first but just outside the current-sheet,
&2 = Hj + 4 TT $,
where </> is the current-function at that point of the sheet.
The value of the component of magnetic force normal to the
sheet is continuous, being the same on both sides of the sheet.
The component of the magnetic force parallel to the current-lines
is also continuous, but the tangential component perpendicular to
the current-lines is discontinuous at the sheet. If s is the length
of a curve drawn on the sheet, the component of magnetic force
T
in the direction of ds is, for the negative side, —T^J and for the
2
positive side, — =-^ — —^ + 4 -n -f •
ds ds ds
The component of the magnetic force on the positive side there
fore exceeds that on the negative side by 4 TT -~ - At a given point
ds
this quantity will be a maximum when ds is perpendicular to the
current-lines.
On the Induction of Electric Currents in a Sheet of Infinite
Conductivity.
654.] It was shewn in Art. 579 that in any circuit
where E is the impressed electromotive force, p the electrokinetic
momentum of the circuit, R the resistance of the circuit, and i the
current round it. If there is no impressed electromotive force and
no resistance, then ~ = 0, or p is constant.
tit
656.] PLANE SHEET. 263
Now 7;, the electrokinetic momentum of the circuit, was shewn
in Art. 588 to be measured by the surface-integral of magnetic
induction through the circuit. Hence, in the case of a current-
sheet of no resistance, the surface-integral of magnetic induction
through any closed curve drawn on the surface must be constant,
and this implies that the normal component of magnetic induction
remains constant at every point of the current-sheet.
655.] If, therefore, by the motion of magnets or variations of
currents in the neighbourhood, the magnetic field is in any way
altered, electric currents will be set up in the current-sheet, such
that their magnetic effect, combined with that of the magnets or
currents in the field, will maintain the normal component of mag
netic induction at every point of the sheet unchanged. If at first
there is no magnetic action, and no currents in the sheet, then
the normal component of magnetic induction will always be zero
at every point of the sheet.
The sheet may therefore be regarded as impervious to magnetic
induction, and the lines of magnetic induction will be deflected by
the sheet exactly in the same way as the lines of flow of an electric
current in an infinite and uniform conducting mass would be
deflected by the introduction of a sheet of the same form made
of a substance of infinite resistance.
If the sheet forms a closed or an infinite surface, no magnetic
actions which may take place on one side of the sheet will produce
any magnetic effect on the other side.
Theory of a Plane Current-sJieet.
656.] We have seen that the external magnetic action of a
current-sheet is equivalent to that of a magnetic shell whose strength
at any point is numerically equal to c/>, the current-function. When
the sheet is a plane one, we may express all the quantities required
for the determination of electromagnetic effects in terms of a single
function, P, which is the potential due to a sheet of imaginary
matter spread over the plane with a surface-density <£. The value
of P is of course r (*&
< (1)
where r is the distance from the point (x, y, z] for which P is cal
culated, to the point x ", y ', 0 in the plane of the sheet, at which the
element dx' dif is taken.
To find the magnetic potential, we may regard the magnetic
264 CURRENT -SHEETS. [657.
shell as consisting of two surfaces parallel to the plane of xy, the
first, whose equation is z = J <?, having1 the surface-density — , and
c
the second, whose equation is z =—\c, having the surface-density
c
The potentials due to these surfaces will be
-P/ c\ and -- P/ cv-
c (*-g) c (*+?)
ft
respectively, where the suffixes indicate that z -- is put for z
s*
in the first expression, and z 4- - for z in the second. Expanding
2i
these expressions by Taylor's Theorem, adding them, and then
making c infinitely small, we obtain for the magnetic potential due
to the sheet at any point external to it,
657.] The quantity P is symmetrical with respect to the plane of
the sheet, and is therefore the same when —z is substituted for z.
H, the magnetic potential, changes sign when — z is put for z.
At the positive surface of the sheet
11 = -— = 2770. (3)
dz
At the negative surface of the sheet
a = -df- = -2v<t>. (4)
CIZ
Within the sheet, if its magnetic effects arise from the magneti
zation of its substance, the magnetic potential varies continu
ously from 2ir<p at the positive surface to — 2ir(p at the negative
surface.
If the sheet contains electric currents, the magnetic force
within it does not satisfy the condition of having a potential.
The magnetic force within the sheet is, however, perfectly deter
minate.
The normal component,
is the same on both sides of the sheet and throughout its sub
stance.
If a and ft be the components of the magnetic force parallel to
657.] VECTOR-POTENTIAL. 265
x and to y at the positive surface, and a, j3' those on the negative
surface dd> /<,%
a =— 27T-^ = — a', (6)
Within the sheet the components vary continuously from a and
/3 to a' and /3'.
The equations -5 j— — —
dii dz
i/
= _^, (8)
dz dx dy
,7 /~] 3 TJ! s7 (~\
(v \JT tt Jj Cu \L
dx dy dz ' j
which connect the components F, G, H of the vector-potential due
to the current-sheet with the scalar potential 12, are satisfied if
we make dP dP
£ — -j- > Cr = =- , JLL = 0. (9)
dy dx
We may also obtain these values by direct integration, thus for F,
Since the integration is to he estimated over the infinite plane
sheet, and since the first term vanishes at infinity, the expression is
reduced to the second term ; and by substituting
d I d \
- -- tor — -j-? - ,
dy r ay r
and remembering that (/> depends on xf and yf ^ and not on HP, y, zt
If H' is the magnetic potential due to any magnetic or electric
system external to the sheet, we may write
F=-J&'dz, (10)
and we shall then have
for the components of the vector-potential due to this system.
266 CURRENT- SHEETS. [658.
658.] Let us now determine the electromotive force at any point
of the sheet, supposing the sheet fixed.
Let X and Zbe the components of the electromotive force parallel
to x and to y respectively, then, by Art. 598, we have
If the electric resistance of the sheet is uniform and equal to &,
X = au, Y = (TV, (14)
where u and v are the components of the current, and if <£ is the
current-function, ^<f> ^
u = -f-t v =— ~ • (15)
dy dx
But, by equation (3),
at the positive surface of the current-sheet. Hence, equations (12)
and (13) may be written
t (16)
dy at
* d+ , .
c j~
where the values of the expressions are those corresponding to the
positive surface of the sheet.
If we differentiate the first of these equations with respect to x,
and the second with respect to ^, and add the results, we obtain
The only value of \jf which satisfies this equation, and is finite
and continuous at every point of the plane, and vanishes at an
infinite distance, is ^ _ Q (19)
Hence the induction of electric currents in an infinite plane sheet
of uniform conductivity is not accompanied with differences of
electric potential in different parts of the sheet.
Substituting this value of ^, and integrating equations (16),
(17), we obtain ^ dP dP ciP'
Since the values of the currents in the sheet are found by
66O.] DECAY OF CURRENTS IN THE SHEET. 267
differentiating1 with respect to as or y, the arbitrary function of z
and t will disappear. We shall therefore leave it out of account.
If we also write for — , the single symbol R, which represents
277
a certain velocity, the equation between P and P' becomes
4f-f+f- w
659.] Let us first suppose that there is no external magnetic
system acting on the current sheet. We may therefore suppose
P/ = 0. The case then becomes that of a system of electric currents
in the sheet left to themselves, but acting on one another by their
mutual induction, and at the same time losing their energy on
account of the resistance of the sheet. The result is expressed
by the equation dP dP
-»"3T = "77 '
dz dt
the solution of which is
P=f(x,y,(z+Rty. (23)
Hence, the value of P on any point on the positive side of the
sheet whose coordinates are x, y> z, and at a time #, is equal to
the value of P at the point #, y, (z + Rt] at the instant when tf=0.
If therefore a system of currents is excited in a uniform plane
sheet of infinite extent and then left to itself, its magnetic effect
at any point on the positive side of the sheet will be the same
as if the system of currents had been maintained constant in the
sheet, and the sheet moved in the direction of a normal from its
negative side with the constant velocity R. The diminution of
the electromagnetic forces, which arises from a decay of the currents
in the real case, is accurately represented by the diminution of the
force on account of the increasing distance in the imaginary case.
660.] Integrating equation (21) with respect to t, we obtain
If we suppose that at first P and P' are both zero, and that a
magnet or electromagnet is suddenly magnetized or brought from
an infinite distance, so as to change the value of P' suddenly from
zero to P', then, since the time-integral in the second member of
(24) vanishes with the time, we must have at the first instant
P = -P'
at the surface of the sheet.
Hence, the system of currents excited in the sheet by the sudden
268 CURRENT -SHEETS. [66 1.
introduction of the system to which Pf is due is such that at the
surface of the sheet it exactly neutralizes the magnetic effect of
this system.
At the surface of the sheet, therefore, and consequently at all
points on the negative side of it, the initial system of currents
produces an effect exactly equal and opposite to that of the
magnetic system on the positive side. We may express this by
saying that the effect of the currents is equivalent to that of an
image of the magnetic system, coinciding in position with that
system, but opposite as regards the direction of its magnetization
and of its electric currents. Such an image is called a negative
image.
The effect of the currents in the sheet on a point on the positive
side of it is equivalent to that of a positive image of the magnetic
system on the negative side of the sheet, the lines joining corre
sponding points being bisected at right angles by the sheet.
The action at a point on either side of the sheet, due to the
currents in the sheet, may therefore be regarded as due to an
image of the magnetic system on the side of the sheet opposite
to the point, this image being a positive or a negative image
according as the point is on the positive or the negative side of
the sheet.
661.] If the sheet is of infinite conductivity, R = 0, and the
second term of (24) is zero, so that the image will represent the
effect of the currents in the sheet at any time.
In the case of a real sheet, the resistance R has some finite value.
The image just described will therefore represent the effect of the
currents only during the first instant after the sudden introduction
of the magnetic system. The currents will immediately begin to
decay, and the effect of this decay will be accurately represented if
we suppose the two images to move from their original positions, in
the direction of normals drawn from the sheet, with the constant
velocity R.
662.] We are now prepared to investigate the system of currents
induced in the sheet by any system, M, of magnets or electro
magnets on the positive side of the sheet, the position and strength
of which vary in any manner.
Let P', as before, be the function from which the direct action
of this system is to be deduced by the equations (3), (9), &c.,
dp
then —j- b t will be the function corresponding to the system re-
664.] MOVING TKAIL OP IMAGES. 269
presented by -=— 8 1. This quantity, which is the increment of M
(it
in the time bt, may be regardejl as itself representing a magnetic
system.
If we suppose that at the time t a positive image of the system
-— r b t is formed on the negative side of the sheet, the magnetic
clt/
action at any point on the positive side of the sheet due to this
image will be equivalent to that due to the currents in the sheet
excited by the change in M during the first instant after the
change, and the image will continue to be equivalent to the
currents in the sheet, if, as soon as it is formed, it begins to move
in the negative direction of z with the constant velocity E.
If we suppose that in every successive element of the time an
image of this kind is formed, and that as soon as it is formed
it begins to move away from the sheet with velocity E, we shall
obtain the conception of a trail of images, the last of which is
in process of formation, while all the rest are moving like a rigid
body away from the sheet with velocity E.
663.] If P/ denotes any function whatever arising from the
action of the magnetic system, we may find P, the corresponding
function arising from the currents in the sheet, by the following
process, which is merely the symbolical expression for the theory
of the trail of images.
Let PT denote the value of P (the function arising from the
currents in the sheet) at the point (x^y, z + Er], and at the time
t — T, and let P'T denote the value of P' (the function arising from
the magnetic system) at the point (#, y, — (z-\-E,r})} and at the
time*-T. Then dPr ^dPT dPT
-=— = JK-j --- T^J [251
dr dz dt
and equation (21) becomes
dP, d^
lh=^u' (26)
and we obtain by integrating with respect to T from r = 0 to r = oo,
as the value of the function P, whence we obtain all the properties
of the current sheet by differentiation, as in equations (3), (9), &c.
664.] As an example of the process here indicated, let us take
the case of a single magnetic pole of strength unity, moving with
uniform velocity in a straight line.
270 CURRENT-SHEETS. [665.
Let the coordinates of the pole at the time t be
The coordinates of the image of the pole formed at the time
t — T are
£=U(*-T), 17 = 0, ^-(c + ttJtf-Tj + tfT),
and if r is the distance of this image from the point (a?, y, z),
To obtain the potential due to the trail of images we have to
calculate d r°° dr
7/7 70 7"
If we write Q2 = u2 4- (R- U>)2,
dr 1
the value of r in this expression being found by making r = 0.
Differentiating this expression with respect to t, and putting
t = 0, we obtain the magnetic potential due to the trail of images,
" ~Q
By differentiating this expression with respect to x or 2, we
obtain the components parallel to x or 0 respectively of the mag
netic force at any point, and by putting x = 0, z = c, and r — 2c
in these expressions, we obtain the following values of the com
ponents of the force acting on the moving pole itself,
665.] In these expressions we must remember that the motion
is supposed to have been going on for an infinite time before the
time considered. Hence we must not take n> a positive quantity,
for in that case the pole must have passed through the sheet
within a finite time.
If we make u = 0, and ft) negative, X = 0, and
z- -1:
"^ * 9
or the pole as it approaches the sheet is repelled from it.
If we make n> = 0, we find Q2 = u
Y-
'
668.] FOKCE ON MOVING POLE. 271
The component X represents a retarding force acting on the pole
in the direction opposite to that of its own motion. For a given
value of R, X is a maximum when u == 1.2772.
When the sheet is a non-conductor, R = oo and X = 0.
When the sheet is a perfect conductor, R = 0 and X = 0.
The component Z represents a repulsion of the pole from the
sheet. It increases as the velocity increases, and ultimately becomes
— - when the velocity is infinite. It has the same value when
R is zero.
666.] When the magnetic pole moves in a curve parallel to the
sheet, the calculation becomes more complicated, but it is easy to
see that the effect of the nearest portion of the trail of images
is to produce a force acting on the pole in the direction opposite
to that of its motion. The effect of the portion of the trail im
mediately behind this is of the same kind as that of a magnet
with its axis parallel to the direction of motion of the pole at
some time before. Since the nearest pole of this magnet is of the
same name with the moving pole, the force will consist partly of
a repulsion, and partly of a force parallel to the former direction
of motion, but backwards. This may be resolved into a retarding
force, and a force towards the concave side of the path of the
moving pole.
667.] Our investigation does not enable us to solve the case
in which the system of currents cannot be completely formed,
on account of a discontinuity or boundary of the conducting
sheet.
It is easy to see, however, that if the pole is moving parallel
to the edge of the sheet, the currents on the side next the edge
will be enfeebled. Hence the forces due to these currents will
be less, and there will not only be a smaller retarding force, but,
since the repulsive force is least on the side next the edge, the pole
will be attracted towards the edge.
Theory of Arago^s Rotating Disk.
668.] Arago discovered* that a magnet placed near a rotating
metallic disk experiences a force tending to make it follow the
motion of the disk, although when the disk is at rest there is
no action between it and the magnet.
This action of a rotating disk was attributed to a new kind
* Annales de Chimie et de Physique, 1826.
272 CURRENT -SHEETS. [668.
of induced magnetization, till Faraday* explained it by means of
the electric currents induced in the disk on account of its motion
through the field of magnetic force.
To determine the distribution of these induced currents, and
their effect on the magnet, we might make use of the results already
found for a conducting sheet at rest acted on by a moving magnet,
availing ourselves of the method given in Art. 600 for treating the
electromagnetic equations when referred to moving systems of axes.
As this case, however, has a special importance, we shall treat it
in a direct manner, beginning by assuming that the poles of the
magnet are so far from the edge of the disk that the effect of the
limitation of the conducting sheet may be neglected.
Making use of the same notation as in the preceding articles
(656-667), we find for the components of the electromotive force
parallel to x and y respectively,
dy d\js
(1)
a- u = y
dt dx
dx d\lf
(TV = — y-y; -- f"> j
' dt dy J
where y is the resolved part of the magnetic force normal to the
disk.
If we now express u and v in terms of $, the current-function,
,»._**, (2)
dx
and if the disk is rotating about the axis of z with the angular
velocity o>, dy dx
1=.*, Jf.— •* (3)
Substituting these values in equations (1), we find
d<t> dty fA\
<T— !- = ya>#— -y-, (4)
dy dx
d(f) d\jf ,..
— o- -—- = y a) y -- f- - (5)
dx * J dy
Multiplying (4) by x and (5) by y} and adding, we obtain
Multiplying (4) by y and (5) by — x, and adding, we obtain
f d<b dfh^ d\ls d\b
*(x-r- +y-r-} = ®-r- -V-r-'
V dx * dy ' dy J dx
«/ «/
* Exp. Res., 81.
668.] ARAGO'S DISK. 273
If we now express these equations in terms of r and 0, where
x — r cos d} y = r sin 6, (8)
they become a ~ = y o> r2 — r -^- > (9)
du dr
(10)
Equation (10) is satisfied if we assume any arbitrary function
of r and 0, and make d
* = arTr-
Substituting- these values in equation (9), it becomes
Dividing by ar2, and restoring the coordinates SB and ^, this
becomes d\ d*x _ « /i 4\
^ + d/ -<ry'
This is the fundamental equation of the theory, and expresses the
relation between the function, x, and the component, y, of the mag
netic force resolved normal to the disk.
Let Q be the potential, at any point on the positive side of the
disk, due to imaginary matter distributed over the disk with the
surface-density x«
At the positive surface of the disk
Hence the first member of equation ( 1 4) becomes
dx2 dy2 ~ 2 77 dz
*S iS
But since Q satisfies Laplace's equation at all points external
to the disk, d20. d20. d20 ,17)
dz*
and equation (14) becomes
j = coy.
2 TT dz*
Again, since Q is the potential due to the distribution x> the
potential due to the distribution $, or -^ , will be •— . From this
du clQ
we obtain for the magnetic potential due to the currents in the disk,
VOL. II.
274 CURRENT- SHEETS. [669.
and for the component of the magnetic force normal to the disk
due to the currents,
*._»»*«.. (20)
71 dz dedz*
If f22 is the magnetic potential due to external magnets, and
if we write r
(21)
the component of the magnetic force normal to the disk due to
the magnets will be
We may now write equation (18), remembering that
y
<r d*Q
Integrating twice with respect to z, and writing R for -— ,
2i TT
(24)
If the values of P and Q are expressed in terms of r, 6, and £
where 7?
f=*--0, (25)
0)
equation (24) becomes, by integration with respect to (,
(2G)
669.] The form of this expression shews that the magnetic action
of the currents in the disk is equivalent to that of a trail of images
of the magnetic system in the form of a helix.
If the magnetic system consists of a single magnetic pole of
strength unity, the helix will lie on the cylinder whose axis is
that of the disk, and which passes through the magnetic pole.
The helix will begin at the position of the optical image of the
pole in the disk. The distance, parallel to the axis between con-
71
secutive coils of the helix, will be 2 IT — . The magnetic effect of
CO
the trail will be the same as if this helix had been magnetized
everywhere in the direction of a tangent to the cylinder perpen
dicular to its axis, with an intensity such that the magnetic moment
of any small portion is numerically equal to the length of its pro
jection on the disk.
670.] SPHERICAL SHEET. 275
The calculation of the effect on the magnetic pole would be
complicated, but it is easy to see that it will consist of —
(1) A dragging force, parallel to the direction of motion of
the disk.
(2) A repulsive force acting from the disk.
(3) A force towards the axis of the disk.
When the pole is near the edge of the disk, the third of these
forces may be overcome by the force towards the edge of the disk,
indicated in Art. 667.
All these forces were observed by Arago, and described by him in
the Annales cle C/iimie for 1826. See also Felici, in Tortolinr's
Annals, iv, p. 173 (1853), and v. p. 35 ; and E. Jochmann, in Crelle's
Journal, Ixiii, pp. 158 and 329; and Pogg. Ann. cxxii, p. 214
(1864). In the latter paper the equations necessary for deter
mining the induction of the currents on themselves are given, but
this part of the action is omitted in the subsequent calculation of
results. The method of images given here was published in the
Proceedings of the Eoyal Society for Feb. 15, 1872.
Spherical Current- Sheet.
670.] Let $ be the current-function at any point Q of a spherical
current-sheet, and let P be the po
tential at a given point, due to a
sheet of imaginary matter distributed
over the sphere with surface-density
<p, it is required to find the magnetic
potential and the vector-potential of
the current-sheet in terms of P.
Let a denote the radius of the
sphere, r the distance of the given
point from the centre, and p the
reciprocal of the distance of the given point from the point Q on
the sphere at which the current-function is (p.
The action of the current-sheet at any point not in its substance
is identical with that of a magnetic shell whose strength at any
point is numerically equal to the current-function.
The mutual potential of the magnetic shell and a unit pole placed
at the point P is, by Art. 410,
T 2
276 CURllENT- SHEETS.
Since p is a homogeneous function of the degree — 1 mr and a,
dp dp
a-/- +r-f = —p,
da dr
Since r and a are constant during the surface-integration,
But if P is the potential due to a sheet of imaginary matter
of surface-density $,
and 12, the magnetic potential of the current-sheet, may be expressed
in terms of P in the form
a=_l-i(P,).
a dr v
671.] We may determine F, the ^-component of the vector-
potential, from the expression given in Art. 416,
where f , ry, f are the coordinates of the element dS, and I, m, n are
the direction-cosines of the normal.
Since the sheet is a sphere, the direction-cosines of the normal are
dp . N o ^
and ^ = (y-,)y = -^,
sothat «_«*=«---
_z dp y dp m
a dy a dz '
multiplying by (/> dS, and integrating over the surface of the sphere,
we find z (].p y dp
a dy a dz
672.] FIELD OF UNIFORM FORCE. 277
x (IP z dP
Similarly G = - -= ---- — 5
a dz a ax
—--—.
a dx a dy
The vector S(, wliose components are F, G, //, is evidently per
pendicular to the radius vector r, and to the vector whose com-
dP dP , dP TC
ponents are -7- > —=- . and -=- . It we determine the lines 01 inter-
dx ay dz
sections of the spherical surface whose radius is r, with the series of
equipotential surfaces corresponding1 to values of P in arithmetical
progression, these lines will indicate by their direction the direction
of §[, and by their proximity the magnitude of this vector.
In the language of Quaternions,
21 = -7PVP.
a
672.] If we assume as the value of P within the sphere
where Yi is a spherical harmonic of degree i, then outside the sphere
The current-function <£ is
2i+l 1
0 = AX A.
47T tf
The magnetic potential within the sphere is
and outside & = i - A ( - ) Y, .
a \r'
For example, let it be required to produce, by means of a wire
coiled into the form of a spherical shell, a uniform magnetic force
M within the shell. The magnetic potential within the shell is, in
this case, a solid harmonic of the first degree of the form
12, — Mr cos 0,
where M is the magnetic force. Hence A = — ^ «2J/, and
d> = — Ma cos 0.
Sir
The current-function is therefore proportional to the distance
from the equatorial plane of the sphere, and therefore the number
of windings of the wire between any two small circles must be
proportional to the distance between the planes of these circles.
278 CURRENT-SHEETS. [673.
If N is the whole number of windings, and if y is the strength
of the current in each winding,
$ = \ Ny cos 0.
Hence the magnetic force within the coil is
47T Ny
M = -- - •
3 a
673.] Let us next find the method of coiling the wire in order
to produce within the sphere a magnetic potential of the form of a
solid zonal harmonic of the second degree,
Here <£ = -—A (f cos20—
If the whole number of windings is N, the number between the
pole and the polar distance 0 is ^ j^sin20.
The windings are closest at latitude 45°. At the equator the
direction of winding changes, and in the other hemisphere the
windings are in the contrary direction.
Let y be the strength of the current in the wire, then within
the shell 4 77
fl =—
O
Let us now consider a Conductor in the form of a plane closed
curve placed anywhere within the shell with its plane perpendicular
to the axis. To determine its coefficient of induction we have to
find the surface-integral of -=- over the plane bounded by the
clz
curve, putting y = 1.
Now ^
Ar
and -=- = —-= Nz.
dz 5 a2
Hence, if S is the area of the closed curve, its coefficient of in
duction is o —
If the current in this conductor is y, there will be, by Art. 583,
a force Z} urging it in the direction of 0, where
„ ,dM 8
and, since this is independent of x, y, z, the force is the same in
whatever part of the shell the circuit is placed.
674.] The method given by Poisson, and described in Art. 437,
LINEAR CURRENT -FUNCTION. 279
may be applied to current-sheets by substituting- for the body
supposed to be uniformly magnetized in the direction of z with
intensity 7, a current-sheet having the form of its surface, and for
which the current-function is 0 — Xz. (1)
The currents in the sheet will be in planes parallel to that of xy,
and the strength of the current round a slice of thickness dz will be
Idz.
The magnetic potential due to this current-sheet at any point
outside it will be TdV ( .
~ ~dz'
At any point inside the sheet it will be
rlV
a=-4V/*-/^-. (3)
dz
The components of the vector-potential are
F = -Icl^, G = I~, 11=0. (4)
dy dx
These results can be applied to several cases occurring in practice.
675.] (1) A plane electric circuit of any form.
Let V be the potential due to a plane sheet of any form of which
the surface-density is unity, then, if for this sheet we substitute
either a magnetic shell of strength 7 or an electric current of
strength I round its boundary, the values of H and of F, G, H will
be those given above.
(2) For a solid sphere of radius a,
V= — - when r is greater than a, (5)
o T
and 7= ~ (3a2—r2) when r is less than a. (6)
o
Hence, if such a sphere is magnetized parallel to z with intensity
7, the magnetic potential will be
H = — I -3 z outside the sphere, (7)
and II = — I z inside the sphere. (8)
«3
If, instead of being magnetized, the sphere is coiled with wire
in equidistant circles, the total strength of current between two
small circles whose planes are at unit distance being 7, then outside
the sphere the value of H is as before, but within the sphere
This is the case already discussed in Art. 672.
280 CURRENT- SHEETS. [676.
(3) The case of an ellipsoid uniformly magnetized parallel to
a given line has been discussed in Art. 437.
If the ellipsoid is coiled with wire in parallel and equidistant
planes, the magnetic force within the ellipsoid will be uniform.
(4) A Cylindric Magnet or Solenoid.
676.] If the body is a cylinder having any form of section and
bounded by planes perpendicular to its generating lines, and
if F! is the potential at the point (a?, y, z) due to a plane area of
surface-density unity coinciding with the positive end of the
solenoid, and Vz the potential at the same point due to a plane area
of surface-density unity coinciding with the negative end, then, if
the cylinder is uniformly and longitudinally magnetized with in
tensity unity, the potential at the point (#,y, z) will be
fi=r1-r2. (10)
If the cylinder, instead of being a magnetized body, is uniformly
lapped with wire, so that there are n windings of wire in unit
of length, and if a current, y, is made to flow through this wire,
the magnetic potential outside the solenoid is as before,
but within the space bounded by the solenoid and its plane ends
12 = ny(47rz + F! — Fg). (12)
The magnetic potential is discontinuous at the plane ends of the
solenoid, but the magnetic force is continuous.
If rlt r2t the distances of the centres of inertia of the positive
and negative plane end respectively from the point (a?, y, z), are
very great compared with the transverse dimensions of the solenoid,
we may write ^_ A v _ A
where A is the area of either section.
The magnetic force outside the solenoid is therefore very small,
and the force inside the solenoid approximates to a force parallel to
the axis in the positive direction and equal to 4 it n y.
If the section of the solenoid is a circle of radius a, the values of
F! and Fg may be expressed in the series of spherical harmonics
given in Thomson and Tait's Natural Philosophy, Art. 546, Ex. II.,
V=2-n\ — rQl + a + ^ — Q2 — — ^<g4-f 1'1'3^Q6+&
when r>a. (15)
6;7-] SOLENOID. 281
In these expressions r is the distance of the point (as, y, z) from
the centre of one of the circular ends of the solenoid, and the zonal
harmonics, Ql, Q2, &c., are those corresponding to the angle 6 which
r makes with the axis of the cylinder.
The first of these expressions is discontinuous when 6 = — , but
2
we must remember that within the solenoid we must add to the
magnetic force deduced from this expression a longitudinal force
4 TT n y.
677.] Let us now consider a solenoid so long that in the part
of space which we consider, the terms depending on the distance
from the ends may be neglected.
The magnetic induction through any closed curve drawn within
the solenoid is 4-nny A', where A is the area of the projection of
the curve on a plane normal to the axis of the solenoid.
If the closed curve is outside the solenoid, then, if it encloses the
solenoid, the magnetic induction through it is 4 TT n y A, where A is
the area of the section of the solenoid. If the closed curve does not
surround the solenoid, the magnetic induction through it is zero.
If a wire be wound ri times round the solenoid, the coefficient of
induction between it and the solenoid is
M— 47rnn'A. (16)
By supposing these windings to coincide with n windings of the
solenoid, we find that the coefficient of self-induction of unit of
length of the solenoid, taken at a sufficient distance from its ex
tremities, is L — 4 Tin2 A. (17)
Near the ends of a solenoid we must take into account the terms
depending on the imaginary distribution of magnetism on the plane
ends of the solenoid. The effect of these terms is to make the co
efficient of induction between the solenoid and a circuit which sur
rounds it less than the value 4^nA} which it has when the circuit
surrounds a very long solenoid at a great distance from either end.
Let us take the case of two circular and coaxal solenoids of the
same length L Let the radius of the outer solenoid be c19 and let
it be wound with wire so as to have % windings in unit of length.
Let the radius of the inner solenoid be c2) and let the number of
windings in unit of length be n2, then the coefficient of induction
between the solenoids, neglecting the effect of the ends, is
M=Gff, (18)
where G = 4 TTW, (19)
and g = TT e£ lnz. (20)
282 CURRENT-SHEETS. [678.
678.] To determine the effect of the positive end of the solenoids
we must calculate the coefficient of induction on the outer solenoid
due to the circular disk which forms the end of the inner solenoid.
For this purpose we take the second expression for V, as given
in equation (15), and differentiate it with respect to r. This gives
the magnetic force in the direction of the radius. We then multiply
this expression by 2 TT r 2 dp, and integrate it with respect to ju, from
pi = 0 to jit = . _ - . This gives the coefficient of induction
V ^2 + Cl2
with respect to a single winding of the outer solenoid at a distance
z from the positive end. We then multiply this by dz, and
integrate with respect to z from z = I to z = 0. Finally, we
multiply the result by % n.2 , and so find the effect of one of the
ends in diminishing the coefficient of induction.
We thus find for the value of the coefficient of mutual induction
between the two cylinders,
M = ±7i2n1nzc22(l—2c1ci), (21)
where r is put, for brevity, for \//2 + c£.
It appears from this, that in calculating the mutual induction of
two coaxal solenoids, we must use in the expression (20) instead of
the true length I the corrected length I — 2 c^ a, in which a portion
equal to ac^ is supposed to be cut off at each end. When the
solenoid is very long compared with its external radius,
(23)
i \
679.] When a solenoid consists of a number of layers of wire of
such a diameter that there are n layers in unit of length, the
number of layers in the thickness dr is n dr, and we have
£
=4 Trfn*dr, and g = TT l\ n2 r2 dr. (24)
If the thickness of the wire is constant, and if the induction take
place between an external coil whose outer and inner radii are x and
y respectively, and an inner coil whose outer and inner radii are
y and z, then, neglecting the effect of the ends,
Gg = $**ln*n*(x-y)(y*-z*). (25)
68o.] INDUCTION COIL. 283
That this may be a maximum, x and z being given, and y
variable, z* , „,
* = *•?-*;*• (2G)
J
This equation gives the best relation between the depths of the
primary and secondary coil for an induction-machine without an
iron core.
If there is an iron core of radius z, then G remains as before, but
g = TT ifn2 (r2 + 4 TT K z2) dr, (27)
-*)- (28)
If y is given, the value of z which gives the maximum value of g is
187TK ,„„,
z = 4 v - • I « " J
3yi87TK+l
When, as in the case of iron, K is a large number, z = f y, nearly.
If we now make x constant, and y and z variable, we obtain the
maximum value of Gg when
x \y\ z : : 4 : 3 : 2. (30)
The coefficient of self-induction of a long solenoid whose outer
and inner radii are x and y> and having a long iron core whose
radius is z, is
L = %7T2ln*(v-y)2(x2 + 2xy + 3y2 + 24;TTKZ2). (31)
680.] We have hitherto supposed the wire to be of uniform
thickness. We shall now determine the law according to which
the thickness must vary in the different layers in order that, for
a given value of the resistance of the primary or the secondary coil,
the value of the coefficient of mutual induction may be a maximum.
Let the resistance of unit of length of a wire, such that n windings
occupy unit of length of the solenoid, be p n2.
The resistance of the whole solenoid is
E = 2iilJ»*rdr. (32)
The condition that, with a given value of R, G may be a maximum
. dG ndR . „ .
is -T- =C~r- , where C is some constant.
*• _ dr l
This gives n2 proportional to - , or the diameter of the wire of
the exterior coil must be proportional to the square root of the
radius.
In order that, for a given value of R, g may be a maximum
*.0, + lS£«.. (33)
284 CURRENT -SHEETS. [68 1.
Hence, if there is no iron core, the diameter of the wire of the
interior coil should be inversely as the square root of the radius,
but if there is a core of iron having a high capacity for magneti
zation, the diameter of the wire should be more nearly directly
proportional to the square root of the radius of the layer.
An Endless Solenoid.
681.] If a solid be generated by the revolution of a plane area A
about an axis in its own plane, not cutting it, it will have the form
of a ring. If this ring be coiled with wire, so that the windings
of the coil are in planes passing through the axis of the ring, then,
if n is the whole number of windings, the current-function of the
layer of wire is $ = — n y 0, where 6 is the angle of azimuth about
the axis of the ring.
If 12, is the magnetic potential inside the ring and 12' that out
side, then 12-12' = 47T(£ + <?= 2ny0 + C.
Outside the ring 12' must satisfy Laplace's equation, and must
vanish at an infinite distance. From the nature of the problem
it must be a function of 0 only. The only value of 12' which fulfils
these conditions is zero. Hence
12' = 0, 12 = 2ny8+C.
The magnetic force at any point within the ring is perpendicular
to the plane passing through the axis, and is equal to 2ny-
where r is the distance from the axis. Outside the ring there is
no magnetic force.
If the form of a closed curve be given by the coordinates z, r,
and 0 of its tracing point as functions of s, its length from a fixed
point, the magnetic induction through the closed curve is
[• z dr
2ny - -j- ds
V0 r ds
taken round the curve, provided the curve is wholly inside the ring.
If the curve lies wholly without the ring, but embraces it, the
magnetic induction through it is
/"' z' dr _ ,
2 n y / — -=-, ds = 2 n y a,
J Q T (IS
where the accented coordinates refer not to the closed curve, but to
a single winding of the solenoid.
The magnetic induction through any closed curve embracing the
68 1.] ENDLESS SOLENOID. 285
ring1 is therefore the same, and equal to 2 n y a, where a is the linear
/*' zf dr'
— -Tjds'. If the closed curve does not embrace the
/ ds
ring, the magnetic induction through it is zero.
Let a second wire be coiled in any manner round the ring, not
necessarily in contact with it, so as to embrace it nf times. The
induction through this wire is 2 n ri y a, and therefore M, the
coefficient of induction of the one coil on the other, is M = 2 n ri a.
Since this is quite independent of the particular form or position
of the second wire, the wires, if traversed by electric currents, will
experience no mechanical force acting between them. By making
the second wire coincide with the first, we obtain for the coefficient
of self-induction of the ring-coil
L = 2 n2 a.
CHAPTER XIII.
PARALLEL CURRENTS.
Cylindrical Conductors.
682.] IN a very important class of electrical arrangements the
current is conducted through round wires of nearly uniform section,
and either straight, or such that the radius of curvature of the axis
of the wire is very great compared with the radius of the transverse
section of the wire. In order to be prepared to deal mathematically
with such arrangements, we shall begin with the case in which the
circuit consists of two very long parallel conductors, with two pieces
joining their ends, and we shall confine our attention to a part of
the circuit which is so far from the ends of the conductors that the
fact of their not being infinitely long does not introduce any
sensible change in the distribution of force.
We shall take the axis of z parallel to the direction of the con
ductors, then, from the symmetry of the arrangements in the part
of the field considered, everything will depend on //, the component
of the vector-potential parallel to z.
The components of magnetic induction become, by equations (A),
m
dH
c — 0.
For the sake of generality we shall suppose the coefficient of
magnetic induction to be p, so that a = /a a, b — /u, /3, where a and (3
are the components of the magnetic force.
The equations (E) of electric currents, Art. GO 7, give
u = 0, v = 0. 4 KW = — — -^ • (3)
dx dy
683.] STRAIGHT WIRE. 287
683.] If the current is a function of r, the distance from the axis
of Zj and if we write
os = r cos 0, and y = r sin 0, (4)
and {3 for the magnetic force, in the direction in which 6 is measured
perpendicular to the plane through the axis of z, we have
4™=f + 10 = 1*08,). (5)
dr r r dr ^
If C is the whole current flowing through a section bounded by
a circle in the plane gey, whose centre is the origin and whose
radius is r, />
<?= / 2trrwdr = %(3r. (6)
JQ
It appears, therefore, that the magnetic force at a given point
due to a current arranged in cylindrical strata, whose common axis
is the axis of z, depends only on the total strength of the current
flowing through the strata which lie between the given point and
the axis, and not on the distribution of the current among the
different cylindrical strata.
For instance, let the conductor be a uniform wire of radius a,
and let the total current through it be C, then, if the current is
uniformly distributed through all parts of the section, w will be
constant, and C=7rwa2'. (7)
The current flowing through a circular section of radius r, r being
less than a, is C'= -nwr2. Hence at any point within the wire,
C
Outside the wire 8 = 2 — . (9)
f
In the substance of the wire there is no magnetic potential, for
within a conductor carrying an electric current the magnetic force
does not fulfil the condition of having a potential.
Outside the wire the magnetic potential is
£l = 2C0. (10)
Let us suppose that instead of a wire the conductor is a metal
tube whose external and internal radii are a-j, and a2, then, if (7 is
the current through the tubular conductor,
C = 7Tw(al2-a.22). (11)
The magnetic force within the tube is zero. In the metal of the
tube, where ;• is between a-^ and a2,
P= 2^-^--2r--2-2, (12)
288 PARALLEL CURRENTS. [684.
and outside the tube, c
/3=2-, (13)
the same as when the current flows through a solid wire.
684.] The magnetic induction at any point is b = p (3, and since,
by equation (2), fi - _ ^ (14)
dr
H^-jppdr. (15)
The value of // outside the tube is
A — 2iJL0Clogr, (16)
where JUQ is the value of /x in the space outside the tube, and A is a
constant, the value of which depends on the position of the return
current.
In the substance of the tube,
a\ ~~ a-2 ai
In the space within the tube H is constant, and
#=^-2MoClog«1 + Me(l + -logr^). (18)
U-^ — U>2 i*^ '
685.] Let the circuit be completed by a return current, flowing
in a tube or wire parallel to the first, the axes of the two currents
being at a distance b. To determine the kinetic energy of the
system we have to calculate the integral
T = \ fjJHw dx cly dz. (19)
If we confine our attention to that part of the system which lies
between two planes perpendicular to the axes of the conductors, and
distant I from each other, the expression becomes
T= \l Hivdxdy. (20)
If we distinguish by an accent the quantities belonging to the
return current, we may write this
^-!-=jJHw'dx'dy'+jJH'wdxcly + jJHwdxdy+jJll'w'dx'dy'. (21)
Since the action of the current on any point outside the tube is
the same as if the same current had been concentrated at the axis
of the tube, the mean value of H for the section of the return
current is A — 2^C log I, and the mean value of H' for the section
of the positive current is A — 2 /u0 GY/ log b.
687.] LONGITUDINAL TENSION. 289
Hence, in the expression for T, the first two terms may be written
AC'-2n()CC'log6) and A'C-2 n0CC'logl>.
Integrating the two latter terms in the ordinary way, and adding
the results, remembering that C+ C' = 0, we obtain the value of
the kinetic energy T. Writing this \LC2, where L is the co
efficient of self-induction of the system of two conductors, we find
as the value of L for unit of length of the system
L
If the conductors are solid wires, a.2 and a<£ are zero, and
T /,2
(23)
aiai
It is only in the case of iron wires that we need take account of
the magnetic induction in calculating their self-induction. In
other cases we may make /x0, /LI, and // all equal to unity. The
smaller the radii of the wires, and the greater the distance between
them, the greater is the self-induction.
To find the Repulsion, X, between the Two Portions of Wire.
686.] By Art. 580 we obtain for the force tending to increase b,
*-ȣ<".
= 2MO|C">, (24)
which agrees with Ampere's formula, when JUQ = 1, as in air.
687.] If the length of the wires is great compared with the
distance between them, we may use the coefficient of self-induction
to determine the tension of the wires arising from the action of the
current.
If Z is this tension,
In one of Ampere's experiments the parallel conductors consist
of two troughs of mercury connected with each other by a floating
bridge of wire. When a current is made to enter at the extremity
of one of the troughs, to flow along it till it reaches one extremity
VOL. II. U
290 PAEALLEL CURRENTS. [688.
of the floating wire, to pass into the other trough through the
floating bridge, and so to return along the second trough, the
floating bridge moves along the troughs so as to lengthen the part
of the mercury traversed by the current.
Professor Tait has simplified the electrical conditions of this
experiment by substituting for the wire a floating siphon of glass
filled with mercury, so that the current flows in mercury through
out its course.
Fig. 40.
This experiment is sometimes adduced to prove that two elements
of a current in the same straight line repel one another, and thus
to shew that Ampere's formula, which indicates such a repulsion
of collinear elements, is more correct than that of Grassmann, which
gives no action between two elements in the same straight line ;
Art. 526.
But it is manifest that since the formulae both of Ampere and of
Grassmann give the same results for closed circuits, and since we
have in the experiment only a closed circuit, no result of the
experiment can favour one more than the other of these theories.
In fact, both formulae lead to the very same value of the
repulsion as that already given, in which it appears that b, the
distance between the parallel conductors is an important element.
When the length of the conductors is not very great compared
with their distance apart, the form of the value of L becomes
somewhat more complicated.
688.] As the distance between the conductors is diminished, the
value of L diminishes. The limit to this diminution is when the
wires are in contact, or when b = al + a2. In this case
fiV (26)
689.] MINIMUM SELF-INDUCTION. 291
This is a minimum when a^ = a2t and then
£ = 2 /(log 4 + 1),
= 2^(1.8863),
= 3.7726^. (27)
This is the smallest value of the self-induction of a round wire
doubled on itself, the whole length of the wire being 2 I.
Since the two parts of the wire must be insulated from each
other, the self-induction can never actually reach this limiting
value. By using broad flat strips of metal instead of round wires
the self-induction may be diminished indefinitely.
On the Electromotive Force required to produce a Current of Varying
Intensity along a Cylindrical Conductor.
689.] When the current in a wire is of varying intensity, the
electromotive force arising from the induction of the current on
itself is different in different parts of the section of the wire, being
in general a function of the distance from the axis of the wire
as well as of the time. If we suppose the cylindrical conductor
to consist of a bundle of wires all forming part of the same circuit,
so that the current is compelled to be of uniform strength in every
part of the section of the bundle, the method of calculation which
we have hitherto used would be strictly applicable. If, however,
we consider the cylindrical conductor as a solid mass in which
electric currents are free to flow in obedience to electromotive force,
the intensity of the current will not be the same at different
distances from the axis of the cylinder, and the electromotive forces
themselves will depend on the distribution of the current in the
different cylindric strata of the wire.
The vector-potential //, the density of the current w, and the
electromotive force at any point, must be considered as functions of
the time and of the distance from the axis of the wire.
The total current, C, through the section of the wire, and the total
electromotive force, JE, acting round the circuit, are to be regarded
as the variables, the relation between which we have to find.
Let us assume as the value of H,
H= S+To + T^+bc. + T.r**, (1)
where S, T0, Tlf &c. are functions of the time.
Then, from the equation
d2H , 1 dH f .
-J-H- H -=- = — 47TW, (2)
dr2 r dr
we find -TIW = Tl + &c + n*TnrZn~2. (3)
U 2
292 PARALLEL CURRENTS. [690.
If p denotes the specific resistance of the substance per unit of
volume, the electromotive force at any point is p w, and this may be
expressed in terms of the electric potential and the vector potential
H by equations (B), Art. 598,
dV dll ,A.
<>w = -^-w
d3> dS dTQ clT^ dTn
-?w = T* + Tt+-W + -WT+^ + ^?T ' (5)
Comparing the coefficients of like powers of r in equations
<s)'nd(5)'
Hence we may write -=- = — — , (9)
T_,dT _£ 1 d'T
J° 2' ^--pTt>- /B"?(i±FaF
690.] To find the total current (7, we must integrate w over the
section of the wire whose radius is a,
ra
C=27T wrdr. (11)
^o
Substituting the value of itw from equation (3), we obtain
(12)
The value of H at any point outside the wire depends only on
the total current C, and not on .the mode in which it is distributed
within the wire. Hence we may assume that the value of H at the
surface of the wire is A C, where A is a constant to be determined
by calculation from the general form of the circuit. Putting H=AC
when r = a, we obtain
2n- (13)
If we now write - = a, a is the value of the conductivity of
P
unit of length of the wire, and we have
(15)
690.] VARIABLE CURRENT. 293
Eliminating T from these two equations, we find
.dC dS, . dC
. = o. (16)
If I is the whole length of the circuit, R its resistance, and E the
electromotive force due to other causes than the induction of the
current on itself, dS E I
Tl=-J' a = K'
dC PcPC P fPC
The first term, RC> of the right-hand member of this equation
expresses the electromotive force required to overcome the resist
ance according to Ohm's law.
The second term, l(A + \)-;- , expresses the electromotive force
dt
which would be employed in increasing the electrokinetic momentum
of the circuit, on the hypothesis that the current is of uniform
strength at every point of the section of the wire.
The remaining terms express the correction of this value, arising
from the fact that the current is not of uniform strength at different
distances from the axis of the wire. The actual system of currents
has a greater degree of freedom than the hypothetical system,
in which the current is constrained to be of uniform strength
throughout the section. Hence the electromotive force required
to produce a rapid change in the strength of the current is some
what less than it would be on this hypothesis.
The relation between the time-integral of the electromotive force
and the time-integral of the current is
(19)
If the current before the beginning of the time has a constant
value C0) and if during the time it rises to the value CL, and re
mains constant at that value, then the terms involving the differ
ential coefficients of C vanish at both limits, and
,\ (20)
the same value of the electromotive impulse as if the current had
been uniform throughout the wire.
294 PARALLEL CURRENTS. [691.
On the Geometrical Mean Distance of Two Figures in a Plane.*
691.] In calculating the electromagnetic action of a current
flowing in a straight conductor of any given section on the current
in a parallel conductor whose section is also given, we have to find
the integral
where doc dy is an element of the area of the first section, dx'dy' an
element of the second section, and r the distance between these
elements, the integration being extended first over every element
of the first section, and then over every element of the second.
If we now determine a line R, such that this integral is equal to
where A1 and A2 are the areas of the two sections, the length of R
will be the same whatever unit of length we adopt, and whatever
system of logarithms we use. If we suppose the sections divided
into elements of equal size, then the logarithm of R, multiplied by
the number of pairs of elements, will be equal to the sum of the
logarithms of the distances of all the pairs of elements. Here R
may be considered as the geometrical mean of all the distances
between pairs of elements. It is evident that the value of R must
be intermediate between the greatest and the least values of r.
If RA and RB are the geometric mean distances of two figures,
A and JB, from a third, C} and if RA+B is that of the sum of the two
figures from C, then
(A + B) log RA+B=A log RA + B log RB.
By means of this relation we can determine R for a compound
figure when we know R for the parts of the figure.
692.] EXAMPLES.
(1) Let R be the mean distance from the point 0 to the line
AB. Let OP be perpendicular to AB, then
AB (log R + 1) = AP log OA + PB log OB+ OP AOB.
i /
Fig. 41.
* Trans. R. S. Edin., 1871-2.
692.]
GEOMETRIC MEAN DISTANCE.
295
(2) For two lines (Fig. 42) of lengths a and b drawn perpendicu
lar to the extremities of a line of length c and on the same side of it.
«£ (2 log 72 +3) = (c2 - (a-b}2) log+/c2 + (a- &)* + c2 log c
4- (a2 — c2) log \/a2 + c2 4- (b2 — c2) log \/b2 4- c2
/ z\ * i a — ^ ~u - • -b
— c(a — o) tan"1 —
Fig. 42.
(3) For two lines, PQ and RS (Fig. 43), whose directions inter
sect at 0.
PQ.RS(2logR+3) = logPR(20P.ORsin20-PR2cosO)
+ logQS(20Q.OSsin20-QS2cosO)
- log PS (2 OP. OS sin2 0 - PS2 cos 0)
-sinO {OP2. SPR- OQ2. S'QR+OR2. PltQ-OS2. PSQ}.
Fig. 43.
(4) For a point 0 and a rectangle ABCD (Fig. 44). Let OP,
OQ, OR, OS, be perpendiculars on the sides, then
AB.AD (2 log 72+ 3) = 2.0P.OQ log OA + 2 .OQ. OR log OB
+ 2. OR. OS log OC + 2.0S.OP logOD
Fig. 44.
296 PARALLEL CURRENTS. [693.
(5) It is not necessary that the two figures should be different, for
we may find the geometric mean of the distances between every pair
of points in the same figure. Thus, for a straight line of length 0,
log 72 = log a—f,
or E = ae~%,
R = 0.223130.
(6) For a rectangle whose sides are a and d,
}0gR = logvV+^-iJiog /y/i + ^-^°g V1 + &
+ ietan-i*+i-tan-i£-«.
o a a b
When the rectangle is a square, whose side is 0,
log 5 = Iog0 + i log 2 + | -ff,
R = 0.447050.
(7) The geometric mean distance of a point from a circular line
is equal to the greater of the two quantities, its distance from the
centre of the circle, and the radius of the circle.
(8) Hence the geometric mean distance of any figure from a
ring bounded by two concentric circles is equal to its geometric
mean distance from the centre if it is entirely outside the ring, but
if it is entirely within the ring
al a2
where 0j and 02 are the outer and inner radii of the ring. R is
in this case independent of the form of the figure within the ring.
(9) The geometric mean distance of all pairs of points in the
ring is found from the equation
log R = ^0! — 2J^ log ^ 4- J *l ~®\ .
For a circular area of radius 0, this becomes
log R = Iog0-i,
or R = ae~*,
R = 0.77880.
For a circular line it becomes
693.] In calculating the coefficient of self-induction of a coil of
uniform section, the radius of curvature being great compared with
693-]
SELF-INDUCTION OF A COIL.
297
the dimensions of the transverse section, we first determine the
geometric mean of the distances of every pair of points of the
section by the method already described, and then we calculate the
coefficient of mutual induction between two linear conductors of
the given form, placed at this distance apart.
This will be the coefficient of self-induction when the total cur
rent in the coil is unity, and the current is uniform at all points of
the section.
But if there are n windings in the coil we must multiply the
coefficient already obtained by n2, and thus we shall obtain the
coefficient of self-induction on the supposition that the windings of
the conducting wire fill the whole section of the coil.
But the wire is cylindric, and is covered with insulating material,
so that the current, instead of being uniformly distributed over the
section, is concentrated in certain parts of it, and this increases the
coefficient of self-induction. Besides this, the currents in the
neighbouring wires have not the same action on the current in a
given wire as a uniformly distributed current.
The corrections arising from these considerations may be de
termined by the method of the geometric mean distance. They
are proportional to the length of the whole wire of the coil, and
may be expressed as numerical quantities, by which we must
multiply the length of the wire in order to obtain the correction
of the coefficient of self-induction.
Let the diameter of the wire be d. It is
covered with insulating material, and wound
into a coil. We shall suppose that the sections
of the wires are in square order, as in Fig. 45,
and that the distance between the axis of each
wire and that of the next is D, whether in
the direction of the breadth or the depth of
the coil. D is evidently greater than d.
We have first to determine th ' excess of
self-induction of unit of length of a cylindric wire of diameter d
over that of unit of length of a square wire of side D, or
, R for the square
Og* R for the circle
o
o
o
o
o
o
o
o
o
Fig. 45.
D
= 2 (log-T + 0.1380606)
298 PARAkCEL CURRENTS. [693.
The inductive action of the eight nearest round wires on the wire
under consideration is less than that of the corresponding eight
square wires on the square wire in the middle by 2x(. 01971).
The corrections for the wires at a greater distance may be neg
lected, and the total correction may be written
2(loge-=- + 0.11835).
The final value of the self-induction is therefore
L — n2M+ 2/(loge -j + 0.11835),
where n is the number of windings, and I the length of the wire,
M the mutual induction of two circuits of the form of the mean
wire of the coil placed at a distance R from each other, where R is
the mean geometric distance between pairs of points of the section.
D is the distance between consecutive wires, and d the diameter
of the wire.
CHAPTER XIV.
CIRCULAR CURRENTS.
Magnetic Potential due to a Circular Current.
694.] THE magnetic potential at a given point, due to a circuit
carrying a unit current, is numerically equal to the solid angle sub
tended by the circuit at that point ; see Arts. 409, 485.
When the circuit is circular, the solid angle is that of a cone
of the second degree, which, when the given point is on the axis
of the circle, becomes a right cone. When the point is not on
the axis, the cone is an elliptic cone, and its solid angle is
numerically equal to the area of the spherical ellipse which it traces
on a sphere whose radius is unity.
This area can be expressed in finite terms by means of elliptic
integrals of the third kind. We shall find it more convenient to
expand it in the form of an infinite series of spherical harmonics, for
the facility with which mathematical operations may be performed
on the general term of such a series z
more than counterbalances the trouble
of calculating a number of terms suffi
cient to ensure practical accuracy.
For the sake of generality we shall
assume the origin at any point on the
axis of the circle, that is to say, on
the line through the centre perpen
dicular to the plane of the circle.
Let 0 (Fig. 46) be the centre of the
circle, C the point on the axis which
we assume as origin, H a point on the
circle.
Describe a sphere with C as centre,
and CH as radius. The circle will lie
on this sphere, and will form a small circle of the sphere of
angular radius a.
Fig. 46.
300 CIRCULAR CURRENTS. [694.
Let CH = c,
OC = b — c cos a,
OH= a = c sin a.
Let A be the pole of the sphere, and Z any point on the axis, and
let CZ=z.
Let R be any point in space, and let CR = r, and ACR = 6.
Let P be the point when CR cuts the sphere.
The magnetic potential due to the circular current is equal to
that due to a magnetic shell of strength unity bounded by the
current. As the form of the surface of the shell is indifferent,
provided it is bounded by the circle, we may suppose it to coincide
with the surface of the sphere.
We have shewn in Art. 670 that if P is the potential due to a
stratum of matter of surface-density unity, spread over the surface
of the sphere within the small circle, the potential due to a mag
netic shell of strength unity and bounded by the same circle is
* = ii(rP).
c dr ^ '
We have in the first place, therefore, to find P.
Let the given point be on the axis of the circle at Z, then the
part of the potential at Z due to an element dS of the spherical
surface at P is $$
~ZP'
This may be expanded in one of the two series of spherical har
monics, r],
or ++&c. + <i + &c
.j>
the first series being convergent when z is less than c, and the
second when z is greater than c.
Writing dS = — c2 dp dfa
and integrating with respect to <£ between the limits 0 and 2?r,
and with respect to //, between the limits cos a and 1, we find
or P=2vQ0dp + to>.+ rQidp. (O
By the characteristic equation of Qi}
695-] SOLID ANGLE SUBTENDED BY A CIRCLE. 301
Hence ^=. (2)
J^ ^ ^(^ + l) dp
This expression fails when i = 0, but since Q0 = 1,
As the function -~ occurs in every part of this investigation we
d //.
shall denote it by the abbreviated symbol Q/. The values of Q/
corresponding to several values of i are given in Art. 698.
We are now able to write down the value of P for any point R,
whether on the axis or not, by substituting r for z, and multiplying
each term by the zonal harmonic of 6 of the same order. For
P must be capable of expansion in a series of zonal harmonics of 0
with proper coefficients. When 0 = 0 each of the zonal harmonics
becomes equal to unity, and the point E lies on the axis. Hence
the coefficients are the terms of the expansion of P for a point on
the axis. We thus obtain the two series
(4)
(4')
695.] We may now find o>, the magnetic potential of the circuit,
by the method of Art. 670, from the equation
We thus obtain the two series
(6)
!C2 i
t? &'(«)& W + &c- + J+
The series (6) is convergent for all values of r less than c, and the
series (6r) is convergent for all values of r greater than <?. At the
surface of the sphere, where r — c, the two series give the same
value for <o when Q is greater than a, that is, for points not
occupied by the magnetic shell, but when 6 is less than a, that is,
at points on the magnetic shell,
0/= CO+47T. (7)
If we assume 0, the centre of the circle, as the origin of co
ordinates, we must put a = - , and the series become
302
CIRCULAR CURRENTS.
1 0 (n a _ 1
-
[696.
. (8)
where the orders of all the harmonics are odd *.
0# the Potential Energy of two Circular Currents.
696.] Let us begin by supposing the two magnetic shells which
are equivalent to the currents to be portions of two concentric spheres,
their radii being c^ and <?2, of which c^ is the greater (Fig. 47).
Let us also suppose that the axes of the two shells coincide, and
that QJ is the angle subtended by
the radius of the first shell, and ez2
the angle subtended by the radius
of the second shell at the centre C.
Let o^ be the potential due to the
first shell at any point within it, then
the work required to carry the second
shell to an infinite distance is the
value of the surface-integral
r/wco,
JJ dr
Hence
Fig. 47.
extended over the second shell.
4** sin* al(y<
or, substituting the value of the integrals from equation (2), Art. 694,
* The value of the solid angle subtended by a circle may be obtained in a more
direct way as follows. —
The solid angle subtended by the circle at the point Z in the axis is easily shewn
i-* (a)) + &c.
C
Expanding this expression in spherical harmonics, we find
(cos a-l) + (Q, ^cosa-Qo (a))- +&c. + (<& (a) coso-
C
for the expansions of cw for points on the axis for which z is less than c or greater
than c respectively. Remembering the equations (42) and (43) of Art. 132 (vol. i.
p. 165), the coefficients in these equations are evidently the same as those we have
now obtained in a more convenient form for computation.
698.] POTENTIAL OF TWO CIRCLES. 303
697.] Let us next suppose that the axis of one of the shells is
turned about C as a centre,, so that it now makes an angle 0 with
the axis of the other shell (Fig. 48). We have only to introduce
the zonal harmonics of 0 into this expression for M, and we find for
the more general value of M,
This is the value of the potential energy due to the mutual
action of two circular currents of unit strength, placed so that
the normals through the centres of the circles meet in a point C
in an angle 0, the distances of the circumferences of the circles from
the point C being <?x and c2 , of which c± is the greater.
If any displacement dx alters the value
of M, then the force acting in the direc
tion of the displacement is X = -=— •
For instance, if the axis of one of the
shells is free to turn about the point C,
so as to cause 0 to vary, then the moment
of the force tending to increase & is 0,
where _ dM
Performing the differentiation, and remembering that
dB
where (j)/ has the same signification as in the former equations,
0 = — 4 7T2 sin2a1 sin2 a2 sin 0 c2 < J — $/(%) §/(a2) Qi(Q) + &c.
^ 1
698.] As t1 e values of Q{ occur frequently in these calculations
the following table of values of the first six degrees may be useful.
In this table /x stands for cos 0, and v for sin 6.
304 CIRCULAR CURRENTS. [699.
699.] It is sometimes convenient to express the series for M in
terms of linear quantities as follows : — •
Let a be the radius of the smaller circuit, I the distance of its
plane from the origin, and c = \/a2-\-b2.
Let A, B, and C be the corresponding- quantities for the larger
circuit.
The series for M may then be written,
A2
M= 1.2.7T2^02COS0
C3
4- 2.3.7T2 -y=- a2b (cos2 6- i sin2(9)
+ 3.4.7T2 A2(£2-*A^ a2 (£2_1 ^2)(COS30_ 3 sin2 fl cog tf)
-f &C.
If we make 0=0, the two circles become parallel and on the
same axis. To determine the attraction between them we may
differentiate M with respect to b. We thus find
' dM
w=*
700.] In calculating the effect of a coil of rectangular section
we have to integrate the expressions already found with respect
to A, the radius of the coil, and _Z?, the distance of its plane from
the origin, and to extend the integration over the breadth and
depth of the coil.
In some cases direct integration is the most convenient, but
there are others in which the following method of approximation
leads to more useful results.
Let P be any function of x and ^, and let it be required to find
the value of P where
T+i* r+4y
Pxy = / / Pdxdy.
J- J -
In this expression P is the mean value of P within the limits of
integration.
Let P0 be the value of P when x = 0 and y = 0, then, expanding
P by Taylor's Theorem,
Integrating this expression between the limits, and dividing the
result by xy> we obtain as the value of P,
7QI.J COIL OF EECT ANGULAR SECTION. 305
In the case of the coil, let the outer and inner radii be A + \ £,
and A — \^ respectively, and let the distance of the planes of the
windings from the origin lie between JB + ^rj and B—\TI, then the
breadth of the coil is r\, and its depth £ these quantities being
small compared with A or C.
In order to calculate the magnetic effect of such a coil we may
write the successive terms of the series as follows :-^-
&C., &c. ;
ft= ™2
= 277^
&c., &c.
The quantities G0, G1, G2, &c. belong to the large coil. The
value of o> at points for which r is less than C is
a, = _27T + 2G0- ^ r Ql (0)- G^r* Q2 ((9)-^-&c.
The quantities gl9 g^ &c. belong to the small coil. The value of
a/ at points for which r is greater than c is
The potential of the one coil with respect to the other when the
total current through the section of each coil is unity is
To find M by Elliptic Integrals.
701.] When the distance of the circumferences of the two circles
VOL. II. *
306 CIRCULAR CURRENTS. [701.
is moderate as compared with the radii of the smaller, the series
already given do not converge rapidly. In every case, however,
we may find the value of M for two parallel circles by elliptic
integrals.
For let b be the length of the line joining the centres of the circles,
and let this line be perpendicular to the planes of the two circles,
and let A and a be the radii of the circles, then
M ••'«»«
/"/"
= / /
the integration being extended round both curves.
In this case,
r2 = A2 + a2 + b2-2Aacos((j>-(l>')
e = — $', ds =
•27T
M
/•
~J
where c ==
and F and E are complete elliptic integrals to modulus c.
From this we get, by differentiating with respect to b and re
membering that c is a function of b,
—c
If fj and /2 denote the greatest and least values of r,
rf =(A + of + V, r* =(A- a)2 + b2,
4*
and if an angle y be taken such that cos y = — ,
where Fy and Ey denote the complete elliptic integrals of the first
and second kind whose modulus is sin y.
If A — a,j cot y = — - , and
i Cb
-^-=
The quantity -^- represents the attraction between two parallel
circular currents, the current in each being unity.
703.] LINES OF MAGNETIC FOKCE. 307
Second Expression for M.
An expression for M, which is sometimes more convenient, is got
by making ^ = — - - , in which case
ri + r2
M = 4
To draw the Lines of Magnetic Force for a Circular Current.
702.] The lines of magnetic force are evidently in planes passing
through the axis of the circle, and in each of these lines the value
of M is constant.
Calculate the value of KQ — ,-= - = — r^ from Legendre's
(/sine — A3in0)
tables for a sufficient number of values of 0.
Draw rectangular axes of so and z on the paper, and, with centre
at the point x = \ a (sin 0 + cosec d), draw a circle with radius
\ a (cosec 0— sin 0). For all points of this circle the value of el will
be sin 0. Hence, for all points of this circle,
= ^ and A =
Now A is the value of x for which the value of M was found.
Hence, if we draw a line for which x = A, it will cut the circle
in two points having the given value of M.
Giving M a series of values in arithmetical progression, the
values of A will be as a series of squares. Drawing therefore a
series of lines parallel to zy for which x has the values found for A,
the points where these lines cut the circle will be the points where
the corresponding lines of force cut the circle.
If we put m = 4 a a, and M = nm, then
A — x = n2Ke a.
We may call n the index of the line of force.
The forms of these lines are given in Fig. XVIII at the end of
this volume. They are copied from a drawing given by Sir W.
Thomson in his paper on ' Vortex Motion*.'
703.] If the position of a circle having a given axis is regarded
as defined by 6, the distance of its centre from a fixed point on
the axis, and «, the radius of the circle, then M, the coefficient
of induction of the circle with respect to any system whatever
* Trans. R. 8.t Edin., vol. xxv. p. 217 (1869).
X 2
308 CIRCULAR CURRENTS. [703.
of magnets or currents, is subject to the following equation
d2M d2M I dM fc x-v
da2 db2 a da
To prove this, let us consider the number of lines of magnetic
force cut by the circle when a or b is made to vary.
(1) Let a become a + ba, b remaining constant. During this
variation the circle, in expanding, sweeps over an annular surface
in its own plane whose breadth is 8 a.
If V is the magnetic potential at any point, and if the axis of y
be parallel to that of the circle, then the magnetic force perpen-
dV
dicular to the plane of the ring- is -7- •
dy
To find the magnetic induction through the annular surface we
have to integrate
where 6 is the angular position of a point on the ring.
But this quantity represents the variation of M due to the
variation of #, or -= — 8 a. Hence
da
dM ^ f2nadT d0 (2]
(2) Let 6 become 6 + 85, a remaining constant. During this
variation the circle sweeps over a cylindric surface of radius a and
length 8£.
The magnetic force perpendicular to this surface at any point is
-s- where r is the distance from the axis. Hence
dr
dM PIT dV JQ ...
— — = — / a-j-dB. (3)
db JQ dr
Differentiating equation (2) with respect to a, and (3) with
respect to I, we get
d*M P«d7 7 f*« dzY .
-— - = / -j-dO+l a-—rde, (4)
da2 JQ dy J0 dr dy
d*M r*' d^v .... ( .
-—— = — a-f-j-dB, (5)
oar J0 dr dy
-j
Hence —^ + -— - = / -j-dO, (6)
da* db2 JQ dy
\dM
= a-da-^y^
Transposing the last term we obtain equation (1).
704.] TWO PARALLEL CIRCLES. 309
Coefficient of Induction of Two Parallel Circles when the Distance
betiveen the Arcs is Small compared with the Hadlus of either
Circle.
704.] We might deduce the value of M in this case from the
expansion of the elliptic integral already given when its modulus
is nearly unity. The following method, however, is a more direct
application of electrical principles.
First Approximation.
Let A and a be the radii of the circles, and b the distance between
their planes, then the shortest distance
between the arcs is
We have to find M19 the magnetic
induction through the circle A, due to a
unit current in a on the supposition that
r is small compared with A or a.
We shall begin by calculating the
magnetic induction through a circle in
the plane of a whose radius is a — c, c being a quantity small com
pared with a (Fig. 49).
Consider a small element ds of the circle a. At a point' in the
plane of the circle, distant p from the middle of ds, measured in
a direction making an angle 6 with the direction of ds, the magnetic
force due to ds is perpendicular to the plane, and equal to
—s sin 6 ds.
P2
If we now calculate the surface-integral of this force over the
space which lies within the circle a, but outside of a circle whose
centre is ds and whose radius is c, we find it
*2asin0 j
— g sin 6 ds d0 dp = {log 8 a — log c — 2} ds.
If c is small, the surface-integral for the part of the annular space
outside the small circle c may be neglected.
We then find for the induction through the circle whose radius
is a— c, by integrating with respect to ds,
Mac = ^ -n a (logStf— logc — 2},
provided c is very small compared with a.
Since the magnetic force at any point, the distance of which
from a curved wire is small compared with the radius of curvature,
/*JT /*
/ /
J J c
310 CIRCULAR CURRENTS. [705.
is nearly the same as if the wire had been straight, we can calculate
the difference between the induction through the circle whose
radius is a — c, and the circle A by the formula
MaA—Mac = 4: 7t a {logo— log r}.
Hence we find the value of the induction between A and a to be
MAa = 4 77 a (log 8 a— log r— 2)
approximately, provided r is small compared with a.
705.] Since the mutual induction between two windings of the
same coil is a very important quantity in the calculation of ex
perimental results, I shall now describe a method by which the
approximation to the value of M for this case can be carried to any
required degree of accuracy.
We shall assume that the value of M is of the form
1 j . j vu j / ri j **/ A f w rf n
where A = a -f ^i# + A2 — \- A2- — \-A3-^-}-A3 -~ + &c.,
a a, a* a*
and B = —2a + B,uo+B9 — + B'^- + B^ + B»°^ +&c.,
2 # 2 & 3 «2 3 «2
where « and « + o? are the radii of the circles, and y the distance
between their planes.
We ; have to determine the values of the coefficients A and B.
It is manifest that only even powers of y can occur in these quan
tities, because, if the sign of y is reversed, the value of M must
remain the same.
We get another set of conditions from the reciprocal property
of the coefficient of induction, which remains the same whichever
circle we take as the primary circuit. The value of M must there
fore remain the same when we substitute a + % for a, and —a? for a?
in the above expression.
We thus find the following conditions of reciprocity by equating
the coefficients of similar combinations of x and y,
A . A A 7?__JLJL
^3 - """^2 ~~^3> -°3 — 3 ~ 2 -<
7°6.]
COIL OF MAXIMUM SELF-INDUCTION.
311
From the general equation of M, Art. 703,
d2M d*M 1 dM
dx2 dy* a + x dx
we obtain another set of conditions,
O // I O J' .. A
2 l" *^ 2 ~"~™ ~^1 3
2 + 2A'
'= 2A
2
&c.;
4 A2+ Al =
Solving these equations and substituting the values of the co
efficients, the series for If becomes
M —
log
+ &C.J
O ^^ 1. I
— ^5 — 2 -p
+ &c
]
To find the form of a coil for which the coefficient of self-in
duction is a maximum, the total length and thickness of the
wire being given.
706.] Omitting the corrections of Art. 705, we find by Art. 673
where n is the number of windings of the wire, a is the mean
radius of the coil, and R is the geometrical mean distance of the
transverse section of the coil from itself. See Art. 690. If this
section is always similar to itself, R is proportional to its linear
dimensions, and n varies as Rz.
Since the total length of the wire is 2 TT an, a varies inversely
as n. Hence
dn _ dR , da dR
- = 2-^-, and — = — 2 -^- ,
n R a R
and we find the condition that L may be a maximum
312 CIRCULAR CURRENTS. [7°6-
If the transverse section of the coil is circular, of radius <?, then,
by Art. 6 9 2, R
Iog7=-i,
and log — = ^,
whence a = 3.22 c ;
or, the mean radius of the coil should be 3.22 times the radius of
the transverse section of the coil in order that such a coil may have
the greatest coefficient of self-induction. This result was found by
Gauss *.
If the channel in which the coil is wound has a square transverse
section, the mean diameter of the coil should be 3.7 times the side
of the square section.
* Werl-e, Gottingen edition, 1867, vol. v. p. 622.
CHAPTER XV.
ELECTROMAGNETIC INSTRUMENTS.
Galvanometers.
707.] A GALVANOMETER is an instrument by means of which an
electric current is indicated or measured by its magnetic action.
When the instrument is intended to indicate the existence of a
feeble current, it is called a Sensitive Galvanometer.
When it is intended to measure a current with the greatest
accuracy in terms of standard units, it is called a Standard Galva
nometer.
All galvanometers are founded on the principle of Schweigger's
Multiplier, in which the current is made to pass through a wire,
which is coiled so as to pass many times round an open space,
within which a magnet is suspended, so as to produce within this
space an electromagnetic force, the intensity of which is indicated
by the magnet.
In sensitive galvanometers the coil is so arranged that its
windings occupy the positions in which their influence on the
magnet is greatest. They are therefore packed closely together
in order to be near the magnet.
Standard galvanometers are constructed so that the dimensions
and relative positions of all their fixed parts may be accurately
known, and that any small uncertainty about the position of the
moveable parts may introduce the smallest possible error into the
calculations.
In constructing a sensitive galvanometer we aim at making the
field of electromagnetic force in which the magnet is suspended as
intense as possible. In designing a standard galvanometer we
wish to make the field of electromagnetic force near the magnet
as uniform as possible, and to know its exact intensity in terms
of the strength of the current.
314 ELECTROMAGNETIC INSTRUMENTS. [708.
On Standard Galvanometers.
708.] In a standard galvanometer the strength of the current
has to be determined from the force which it exerts on the sus
pended magnet. Now the distribution of the magnetism within
the magnet, and the position of its centre when suspended, are not
capable of being determined with any great degree of accuracy.
Hence it is necessary that the coil should be arranged so as to
produce a field of force which is very nearly uniform throughout
the whole space occupied by the magnet during its possible motion.
The dimensions of the coil must therefore in general be much larger
than those of the magnet.
By a proper arrangement of several coils the field of force within
them may be made much more uniform than when one coil only
is used, and the dimensions of the instrument may be thus reduced
and its sensibility increased. The errors of the linear measurements,
however, introduce greater uncertainties into the values of the
electrical constants for small instruments than for large ones. It
is therefore best to determine the electrical constants of small
instruments, not by direct measurement of their dimensions, but
by an electrical comparison with a large standard instrument, of
which the dimensions are more accurately known ; see Art. 752.
In all standard galvanometers the coils are circular. The channel
in which the coil is to be wound is carefully turned. Its breadth
Fig. 50.
is made equal to some multiple, n, of the diameter of the covered
wire. A hole is bored in the side of the channel where the wire is
709.] MEASUKEMENT OF THE COIL. 315
to enter, and one end of the covered wire is pushed out through
this hole to form the inner connexion of the coil. The channel is
placed on a lathe, and a wooden axis is fastened to it; see Fig. 50.
The end of a long string is nailed to the wooden axis at the same
part of the circumference as the entrance of the wire. The whole
is then turned round, and the wire is smoothly and regularly laid
on the bottom of the channel till it is completely covered by n
windings. During this process the string has been wound n times
round the wooden axis, and a nail is driven into the string at the
^th turn. The windings of the string should be kept exposed
so that they can easily be counted. The external circumference
of the first layer of windings is then measured and a new layer
is begun, and so on till the proper number of layers has been
wound on. The use of the string is to count the number of
windings. If for any reason we have to unwind part of the coil,
the string is also unwound, so that we do not lose our reckoning
of the actual number of windings of the coil. The nails serve
to distinguish the number of windings in each layer.
The measure of the circumference of each layer furnishes a test
of the regularity of the winding, and enables us to calculate the
electrical constants of the coil. For if we take the arithmetic mean
of the circumferences of the channel and of the outer layer, and
then add to this the circumferences of all the intermediate layers,
and divide the sum by the number of layers, we shall obtain the
mean circumference, and from this we can deduce the mean radius
of the coil. The circumference of each layer may be measured by
means of a steel tape, or better by means of a graduated wheel
which rolls on the coil as the coil revolves in the process of
winding. The value of the divisions of the tape or wheel must
be ascertained by comparison with a straight scale.
709.] The moment of the force with which a unit current in
the coil acts upon the suspended apparatus may be expressed in
the series ^ gin Q + ^ gin Q ^ ^ + &c ^
where the coefficients G refer to the coil, and the coefficients g to
the suspended apparatus, 0 being the angle between the axis of
the coil and that of the suspended apparatus ; see Art. 700.
When the suspended apparatus is a thin uniformly and longitud
inally magnetized bar magnet of length 2 1 and strength unity,
suspended by its middle,
^i = 2^, #2 = 0, #3=2£3, &c.
316 ELECTROMAGNETIC INSTRUMENTS. [7IQ-
The values of the coefficients for a magnet of length 2 1 magnetized
in any other way are smaller than when it is magnetized uni
formly.
710.] When the apparatus is used as a tangent galvanometer,
the coil is fixed with its plane vertical and parallel to the direction
of the earth's magnetic force. The equation of equilibrium of the
magnet is in this case
m^HcosO = my sin0 {6^+ G2$2 Q/^ + fec.},
where mg^ is the magnetic moment of the magnet, .7? the horizontal
component of the terrestrial magnetic force, and y the strength
of the current in the coil. When the length of the magnet is
small compared with the radius of the coil the terms after the first
in G and g may be neglected, and we find
TT
y = -=• cot 0.
Gi
The angle usually measured is the deflexion, b, of the magnet
which is the complement of 0, so that cot 0 = tan 8.
The current is thus proportional to the tangent of the deviation,
and the instrument is therefore called a Tangent Galvanometer.
Another method is to make the whole apparatus moveable about
a vertical axis, and to turn it till the magnet is in equilibrium with
its axis parallel to the plane of the coil. If the angle between the
plane of the coil and the magnetic meridian is 8, the equation of
equilibrium is
&c-l >
whence y = -^ - 5 — .sin 8.
(G^-fec.)
Since the current is measured by the sine of the deviation, the
instrument when used in this way is called a Sine Galvanometer.
The method of sines can be applied only when the current is
so steady that we can regard it as constant during the time of
adjusting the instrument and bringing the magnet to equi
librium.
711.] We have next to consider the arrangement of the coils
of a standard galvanometer.
The simplest form is that in which there is a single coil, and
the magnet is suspended at its centre.
Let A be the mean radius of the coil, £ its depth, rj its breadth,
and n the number of windings, the values of the coefficients are
712.] TANGENT GALVANOMETEE. 317
£4 = 0, &c.
The principal correction is that arising1 from G3. The series
becomes G^ yt ( 1 — | -p ^ (cos2 0 — J sin2 0)) •
V 1
The factor of correction will differ most from unity when the
magnet is uniformly magnetized and when 0 = 0. In this case it
I2
becomes 1 — 2 ~^ • It vanishes when tan 0 = 2, or when the de-
.4
flexion is tan"1 4, or 26°34'. Some observers, therefore, arrange
their experiments so as to make the observed deflexion as near
this angle as possible. The best method, however, is to use a
magnet so short compared with the radius of the coil that the
correction may be altogether neglected.
The suspended magnet is carefully adjusted so that its centre
shall coincide as nearly as possible with the centre of the coil. If,
however, this adjustment is not perfect, and if the coordinates of
the centre of the magnet relative to the centre of the coil are os, y, z,
z being measured parallel to the axis of the coil, the factor of
correction is (l 4- 3 °° ) •
When the radius of the coil is large, and the adjustment of the
magnet carefully made, we may assume that this correction is
insensible.
Gaugavn?* Arrangement.
712.] In order to get rid of the correction depending on G3
Gaugain constructed a galvanometer in which this term was ren
dered zero by suspending the magnet, not at the centre of the
coil, but at a point on the axis at a distance from the centre equal
to half the radius of the coil. The form of G is
and, since in this arrangement B = \ A, G3 = 0.
This arrangement would be an improvement on the first form
if we could be sure that the centre of the suspended magnet is
318 ELECTROMAGNETIC INSTRUMENTS. [713.
exactly at the point thus defined. The position of the centre of the
magnet, however, is always uncertain, and this uncertainty intro
duces a factor of correction of unknown amount depending on G2 and
of the form (l — £ -r) , where z is the unknown excess of distance
^4
of the centre of the magnet from the plane of the coil. This
correction depends on the first power of -j . Hence Gaugain's coil
with eccentrically suspended magnet is subject to far greater un
certainty than the old form.
Helmholtz's Arrangement,
713.] Helmholtz converted Gaugain's galvanometer into a trust
worthy instrument by placing a second coil, equal to the first, at
an equal distance on the other side of the magnet.
By placing the coils symmetrically on both sides of the magnet
we get rid at once of all terms of even order.
Let A be the mean radius of either coil, the distance between
their mean planes is made equal to A^ and the magnet is suspended
at the middle point of their common axis. The coefficients are
& =
G3 = 0.0512 — (31 £2 - 36rj2),
GB= -0.73728
where n denotes the number of windings in both coils together.
It appears from these results that if the section of the coils be
rectangular, the depth being f and the breadth 17, the value of
6r3, as corrected for the finite size of the section, will be small, and
will vanish, if £ is to 77 as 36 to 31.
It is therefore quite unnecessary to attempt to wind the coils
upon a conical surface, as has been done by some instrument makers,
for the conditions may be satisfied by coils of rectangular section,
which can be constructed with far greater accuracy than coils
wound upon an obtuse cone.
The arrangement of the coils in Helmholtz's double galvanometer
is represented in Fig. 54, Art. 725.
715.] GALVANOMETER OF THREE COILS. 319
The field of force due to the double coil is represented in section
in Fig. XIX at the end of this volume.
Galvanometer of Four Coils.
714.] By combining four coils we may get rid of the coefficients
G2, G3, G±, G5, and G6. For by any symmetrical combinations
we get rid of the coefficients of even orders Let the four coils
be parallel circles belonging to the same sphere, corresponding
to angles 6, (j>, TT— <£, and TT — 0.
Let the number of windings on the first and fourth coil be ny
and the number on the second and third pn. Then the condition
that G3 = 0 for the combination gives
ft sin2 0 q; (0) + ^ft sin2 $ Q9' (c/>) = 0, (1)
and the condition that G5= 0 gives
ft sin2 6 <25' (6) + pn sin2 <£ Q/ (<#>) = 0, (2)
Putting sin2 0 = x and sin2 $ = y^ (3)
and expressing Q3' and Q5' (Art. 698) in terms of these quantities,
the equations (1) and (2) become
= 0, (4)
= 0. (5)
Taking twice (4) from (5), and dividing by 3, we get
6#2-7#3 + 6j?y2-7j^3 = 0. (6)
Hence, from (4) and (6),
_ x 5x— 4_ x2 7# — 6
P=y I=5j=/6=7^'
and we obtain
7 a?— 6 32 7x— 6
= f
— 4
Both x and y are the squares of the sines of angles and must
therefore lie between 0 and 1 . Hence, either x is between 0 and -f ,
in which case y is between -f- and 1, and p between co and ^%,
or else x is between f and 1, in which case y is between 0 and
f, and p between 0 and |f.
Galvanometer of Three Colls.
715.] The most convenient arrangement is that in which x = 1.
Two of the coils then coincide and form a great circle of the sphere
whose radius is C. The number of windings in this compound
coil is 64. The other two coils form small circles of the sphere.
The radius of each of them is \/ C. The distance of either of
320 ELECTROMAGNETIC INSTRUMENTS. [715.
them from the plane of the first is \/'i C. The number of windings
on each of these coils is 49.
1 20
The value of G1 is ~-^- <
L>
This arrangement of coils is represented in Fig. 51,
Fig. 51.
Since in this three-coiled galvanometer the first term after G1
which has a finite value is (r7, a large portion of the sphere on
whose surface the coils lie forms a field of force sensibly uniform.
If we could wind the wire over the whole of a spherical surface,
as described in Art. 627, we should obtain a field of perfectly
uniform force. It is practically impossible, however, to distribute
the windings on a spherical surface with sufficient accuracy, even
if such a coil were not liable to the objection that it forms a closed
surface, so that its interior is inaccessible.
By putting the middle coil out of the circuit, and making the
current flow in opposite directions through the two side coils, we
obtain a field of force which exerts a nearly uniform action in
the direction of the axis on a magnet or coil suspended within it,
with its axis coinciding with that of the coils; see Art. 673. For
in this case all the coefficients of odd orders disappear, and since
Hence the expression for the magnetic potential near the centre
of the coil becomes
^ QG W + &C.J
7 1 6.] THICKNESS OF THE WIRE. 321
On the Proper Thickness of the Wire of a Galvanometer, the External
Resistance being given.
716.] Let the form of the channel in which the galvanometer
coil is to be wound be given, and let it be required to determine
whether it ought to be filled with a long thin wire or with a shorter
thick wire.
Let I be the length of the wire, y its radius, y + b the radius
of the wire when covered, p its specific resistance, g the value of
G for unit of length of the wire, and r the part of the resistance
which is independent of the galvanometer.
The resistance of the galvanometer wire is
„ P i
Jt= -- 5 •
ity*
The volume of the coil is
7= 4l(y + b)2.
The electromagnetic force is y G, where y is the strength of the
current and G — gl.
If E is the electromotive force acting in the circuit whose
resistance is R + r, E = y (R + r).
The electromagnetic force due to this electromotive force is
G
which we have to make a maximum by the variation of y and I.
Inverting the fraction, we find that
_P _J_ r
TT<? f gl
is to be made a minimum. Hence
pdy rdl
& - o H — 75— = 0.
7T^3 I2
If the volume of the coil remains constant
dl dy
-y- + 2 -*- = 0.
1 y + 6
Eliminating dl and dy, we obtain
p y + b _ r
r
or
R y
Hence the thickness of the wire of the galvanometer should be
such that the external resistance is to the resistance of the gal
vanometer coil as the diameter of the covered wire to the diameter
of the wire itself.
VOL. IT. Y
322 ELECTROMAGNETIC INSTRUMENTS. [717.
On Sensitive Galvanometers.
717.] In the construction of a sensitive galvanometer the aim
of every part of the arrangement is to produce the greatest possible
deflexion of the magnet by means of a given small electromotive
force acting between the electrodes of the coil.
The current through the wire produces the greatest effect when
it is placed as near as possible to the suspended magnet. The
magnet, however, must be left free to oscillate, and therefore there
is a certain space which must be left empty within the coil. This
defines the internal boundary of the coil.
Outside of this space each winding must be placed so as to have
the greatest possible effect on the magnet. As the number of
windings increases, the most advantageous positions become filled
up, so that at last the increased resistance of a new winding
diminishes the effect of the current in the former windings more
than the new winding itself adds to it. By making the outer
windings of thicker wire than the inner ones we obtain the greatest
magnetic effect from a given electromotive force.
718.] We shall suppose that the windings of the galvanometer
are circles, the axis of the galvanometer passing through the centres
of these circles at right angles to their planes.
Let r sin Q be the radius of one of these circles, and r cos 0 the
distance of its centre from the centre of the galvanometer, then,
if I is the length of a portion of wire coinciding with this circle,
and y the current which flows in it, the
magnetic force at the centre of the gal
vanometer resolved in the direction of
the axis is sin Q
y'-p-
If we write r2 = x2 sin 0, (1)
this expression become^ y —^ •
x
Hence, if a surface be constructed
similar to those represented in section
in Fig. 52, whose polar equation is
r2 = x* sin 0, (2)
where a?x is any constant, a given length
of wire bent into the form of a circular
g ' arc will produce a greater magnetic
effect when it lies within this surface than when it lies outside it.
719.] SENSITIVE GALYANOMETEK. 323
It follows from this that the outer surface of any layer of wire
ought to have a constant value of x, for if x is greater at one place
than another a portion of wire might be transferred from the first
place to the second, so as to increase the force at the centre of the
galvanometer.
The whole force due to the coil is y G, where
G
•n-
the integration being extended over the whole length of the wire,
x being considered as a function of I.
719.] Let y be the radius of the wire, its transverse section will
be 7r^2. Let p be the specific resistance of the material of which
the wire is made referred to unit of volume, then the resistance of a
length I is — ^ } and the whole resistance of the coil is
*f
/* 77
(4)
where y is considered a function of I.
Let Y2 be the area of the quadrilateral whose angles are the
sections of the axes of four neighbouring wires of the coil by a
plane through the axis, then Y2l is the volume occupied in the coil
by a length I of wire together with its insulating covering, and
including any vacant space necessarily left between the windings
of the coil. Hence the whole volume of the coil is
r=jY»dl,
where Y is considered a function of /.
But since the coil is a figure of revolution
V — 2 TT jjr2 sin 0 dr dO, (6)
or, expressing r in terms of x, by equation (2),
V = 2 TT I j a? (sin 0)* dan dB. (7)
Now 27T / (sill 0)$ dO is a numerical quantity, call it JV, then
•'o
where F"0 is the volume of the interior space left for the
magnet.
Let us now consider a layer of the coil contained between the
surfaces x and x + das.
Y 2,
324 ELECTROMAGNETIC INSTRUMENTS. [7J9-
The volume of this layer is
x = Y2dl, (9)
where dl is the length of wire in this layer.
This gives us dl in terms of dx. Substituting this in equations
(3) and (4), we find
where f/(r and f/.S represent the portions of the values of G and of
It due to this layer of the coil.
Now if E be the given electromotive force,
where r is the resistance of the external part of the circuit, in
dependent of the galvanometer, and the force at the centre is
G
si
We have therefore to make -= — a maximum, by properly ad-
JK -\- T
justing the section of the wire in each layer. This also necessarily
involves a variation of Y because Y depends on y.
Let G0 and JRQ be the values of G and of R + r when the given
layer is excluded from the calculation. We have then
R0+dR
and to make this a maximum by the variation of the value of y for
the given layer we must have
£,*«
(13>
.
ay
C1
Since dx is very small and ultimately vanishes, ^- will be sensibly,
**o
and ultimately exactly, the same whichever layer is excluded, and
we may therefore regard it as constant. We have therefore, by (10)
and (11), X2 Y dy. PR + r
f 0 + 7 3r) = 1-^- = constant- (14)
If the method of covering the wire and of winding it is such
that the proportion between the space occupied by the metal of
720.] SENSITIVE GALVANOMETER. 325
the wire bears the same proportion to the space between the wires
whether the wire is thick or thin, then
and we must make both y and Y proportional to x, that is to say,
the diameter of the wire in any layer must be proportional to the
linear dimension of that layer.
If the thickness of the insulating covering is constant and equal
to d, and if the wires are arranged in square order,
Y=2(y + b\ (15)
and the condition is
= constant. (16)
In this case the diameter of the wire increases with the diameter
of the layer of which it forms part, but not in so high a ratio.
If we adopt the first of these two hypotheses, which will be nearly
true if the wire itself nearly fills up the whole space, then we may
put y = ax, Y= $y,
where a and ft are constant numerical quantities, and
where a is a constant depending upon the size and form of the free
space left inside the coil.
Hence, if we make the thickness of the wire vary in the same
ratio as as, we obtain very little advantage by increasing the
external size of the coil after the external dimensions have become
a large multiple of the internal dimensions.
720.] If increase of resistance is not regarded as a defect, as
when the external resistance is far greater than that of the gal
vanometer, or when our only object is to produce a field of intense
force, we may make y and Y constant. We have then
N
G= 71 (*-")>
-p 1 Pf/*.3 n %\
~ 3 Yf> Jj'* * ''
where a is a constant depending on the vacant space inside the
coil. In this case the value of G increases uniformly as the
dimensions of the coil are increased, so that there is no limit to
the value of G except the labour and expense of making the coil.
326
ELECTROMAGNETIC INSTRUMENTS,
[721.
On Suspended Coils.
721.] In the ordinary galvanometer a suspended magnet is acted
on by a fixed coil. But if the coil can be suspended with sufficient
delicacy, we may determine the action of the magnet, or of another
coil on the suspended coil, by its deflexion from the position of
equilibrium.
We cannot, however, introduce the electric current into the coil
unless there is metallic connexion between the electrodes of the
battery and those of the wire of the coil. This connexion may be
made in two different ways, by the Bifilar Suspension, and by wires
in opposite directions.
The bifilar suspension has already been described in Art. 459
as applied to magnets. The arrangement of the upper part of the
suspension is shewn in Fig. 55. When applied to coils, the two
fibres are no longer of silk but of metal, and since the torsion of
a metal wire capable of supporting the coil and transmitting the
current is much greater than that of a silk fibre, it must be taken
specially into account. This suspension has been brought to great
perfection in the instruments constructed by M. Weber.
The other method of suspension is by means of a single wire
which is connected to one extremity of the coil. The other ex
tremity of the coil is connected to another wire which is made
to hang down, in the same vertical straight line with the first wire,
into a cup of mercury, as is shewn in Fig. 57, Art. 729. In certain
cases it is convenient to fasten the extremities of the two wires to
pieces by which they may be tightly stretched, care being taken
that the line of these wires passes
through the centre of gravity of the
coil. The apparatus in this form
may be used when the axis is not
vertical ; see Fig. 53.
722.] The suspended coil may be
used as an exceedingly sensitive gal
vanometer, for, by increasing the in
tensity of the magnetic force in the
field in which it hangs, the force due
to a feeble current in the coil may
be greatly increased without adding
to the mass of the coil. The mag
netic force for this purpose may be
Fig. 53.
produced by means of permanent magnets, or by electromagnets
723-] SUSPENDED COIL. 327
excited by an auxiliary current, and it may be powerfully concen
trated on the suspended coil by means of soft iron armatures. Thus,
in Sir W. Thomson's recording apparatus, Fig. 53, the coil is sus
pended between the opposite poles of the electromagnets N and S,
and in order to concentrate the lines of magnetic force on the ver
tical sides of the coil, a piece of soft iron, 1), is fixed between the
poles of the magnets. This iron becoming magnetized by induc
tion, produces a very powerful field of force, in the intervals between
it and the two magnets, through which the vertical sides of the
coil are free to move, so that the coil, even when the current
through it is very feeble, is acted on by a considerable force
tending to turn it about its vertical axis.
723.] Another application of the suspended coil is to determine,
by comparison with a tangent galvanometer, the horizontal com
ponent of terrestrial magnetism.
The coil is suspended so that it is in stable equilibrium when
its plane is parallel to the magnetic meridian. A current y is
passed through the coil and causes it to be deflected into a new
position of equilibrium, making an angle 0 with the magnetic
meridian. If the suspension is bifilar, the moment of the couple
which produces this deflexion is I1 sin 0, and this must be equal to
HyffcosO, where His the horizontal component of terrestrial mag
netism, y is the current in the coil, and g is the sum of the areas of
all the windings of the coil. Hence
F
II y — — tan0.
g
If A is the moment of inertia of the coil about its axis of sus
pension, and Tthe time of a single vibration,
FT2 = v*A,
Ti^A
and we obtain Hy = - tan 0.
If the same current passes through the coil of a tangent galva
nometer, and deflects the magnet through an angle 0,
y
where G is the principal constant of the tangent galvanometer, Art. 710,
From these two equations we obtain
7T tr /AGkaxid TT /A tan 0 tan rf>
: ~T A/Tte^T' = T V -oT
Tliis method wa^ given by F. Kohlrausch *.
* r"ogg., Ann. cxxxviii, Feb. 18G9.
328 ELECTROMAGNETIC INSTRUMENTS. [?24-
724.] Sir William Thomson has constructed a single instrument
by means of which the observations required to determine H and y
may be made simultaneously by the same observer.
The coil is suspended so as to be in equilibrium with its plane
in the magnetic meridian, and is deflected from this position
when the current flows through it. A very small magnet is sus
pended at the centre of the coil, and is deflected by the current in
the direction opposite to that of the deflexion of the coil. Let the
deflexion of the coil be 6, and that of the magnet 0, then the
energy of the system is
Hy g sm9 + my G sin (0 — fy — Hmcos 0 — Fcos 9.
Differentiating with respect to 0 and 0, we obtain the equa
tions of equilibrium of the coil and of the magnet respectively,
Hyg cos 0 + my (7 cos (0 — 0) + F sin Q = 0,
— my G cos (6 — 0)-f Hm sin 0 = 0.
From these equations we find, by eliminating H or y} a quadratic
equation from which y or // may be found. If m, the magnetic
moment of the suspended mag-net, is very small, we obtain the
following approximate values
j _ IT /— ^<?sin0cos(0 — 0) L mG cos (6 — 0)
' ~T 'V g cos 6 sin 0 2 g cos0
77 / — ^4 sin 0 sin 0 ^m sin0
" ~T V G g cos 6 cos (0—0) ~~ 2 7 cos^ "
In these expressions G and g are the principal electric constants
of the coil, A its moment of inertia, T its time of vibration, m the
magnetic moment of the magnet, H the intensity of the horizontal
magnetic force, y the strength of the current, 0 the deflexion of the
coil, and 0 that of the magnet.
Since the deflexion of the coil is in the opposite direction to the
deflexion of the magnet, these values of H and y will always be
real.
Weber's Electrody nanometer.
725.] In this instrument a small coil is suspended by two wires
within a larger coil which is fixed. When a current is made to
flow through both coils, the suspended coil tends to place itself
parallel to the fixed coil. This tendency is counteracted by the
moment of the forces arising from the bifilar suspension, and it is
also affected by the action of terrestrial magnetism on the sus
pended coil.
725.] ELECTRODYNAMOMETER. 329
In the ordinary use of the instrument the planes of the two coils
are nearly at right angles to each other, so that the mutual action
of the currents in the coils may be as great as possible, and the
plane of the suspended coil is nearly at right angles to the magnetic
meridian, so that the action of terrestrial magnetism may be as
small as possible.
Let the magnetic azimuth of the plane of the fixed coil be a,
and let the angle which the axis of the suspended coil makes with
the plane of the fixed coil be Q + fi, where (3 is the value of this
angle when the coil is in equilibrium and no current is flowing,
and"* 6 is the deflexion due to the current. The equation of equi
librium is
Let us suppose that the instrument is adjusted so that a and j3
are both very small, and that Hgy^ is small compared with F.
We have in this case, approximately,
(r^y1y2cos/3 Zfyy2sin(a-|-/3) HGg^y^y^ G2y2yl2y22smj3
If the deflexions when the signs of yl and y2 are changed are
as follows : e when is , and ,
then we find
F
yl y2 — J — — (tan 0J + tan 02 — tan 03 — tan 04).
If it is the same current which flows through both coils we may put
yl y2 = y2, and thus obtain the value of y.
When the currents are not very constant it is best to adopt this
method, which is called the Method of Tangents.
If the currents are so constant that we can adjust /3, the angle
of the torsion-head of the instrument, we may get rid of the
correction for terrestrial magnetism at once by the method of sines.
In this method /3 is adjusted till the deflexion is zero, so that
0=_/3.
If the signs of y1 and y2 are indicated by the suffixes of /3 as
before,
Fsin & = -Fsin P3 = — Gffyly2 + Hg y2 sin a,
F sin )32 = — ^sin /34 = — Gg yl y^ — Rg y2 sin a,
F
and Yl y2 = - — ^ (sin fa + sin fa - sin fa - sin fa).
330
ELECTROMAGNETIC INSTRUMENTS.
[725<
725.] ELECTRODYNAMOMETER. 331
This is the method adopted by Mr. Latimer Clark in his use
of the instrument constructed by the Electrical Committee of the
British Association. We are indebted to Mr. Clark for the drawing
of the electrodynamometer in Figure 54, in which Helmholtz's
arrangement of two coils is adopted both for the fixed and for the
suspended coil*. The torsion-head of the instrument, by which
the bifilar suspension is adjusted, is represented in Fig. 55. The
Fig. 55.
equality of the tension of the suspension wires is ensured by their
being attached to the extremities of a silk thread which passes over
a wheel, and their distance is regulated by two guide-wheels, which
can be set at the proper distance. The suspended coil can be moved
vertically by means of a screw acting on the suspension-wheel,
and horizontally in two directions by the sliding pieces shewn at
the bottom of Fig. 55. It is adjusted in azimuth by means of the
torsion-screw, which turns the torsion-head round a vertical axis
(see Art. 459). The azimuth of the suspended coil is ascertained
by observing the reflexion of a scale in the mirror, shewn just
beneath the axis of the suspended coil.
* In the actual instrument, the wires conveying the current to and from the coils
are not spread out as displayed in the figure, but are kept as close together as pos
sible, so as to neutralize each other's electromagnetic action.
332 ELECTROMAGNETIC INSTRUMENTS.
The instrument originally constructed by Weber is described in
his Elektroctynamiscke Maasbeslimmungen. It was intended for the
measurement of small currents, and therefore both the fixed and
the suspended coils consisted of many windings, and the suspended
coil occupied a larger part of the space within the fixed coil than in
the instrument of the British Association, which was primarily in
tended as a standard instrument, with which more sensitive instru
ments might be compared. The experiments which he made with
it furnish the most complete experimental proof of the accuracy of
Ampere's formula as applied to closed currents, and form an im
portant part of the researches by which Weber has raised the
numerical determination of electrical quantities to a very high rank
as regards precision.
Weber's form of the electrodynarnometer, in which one coil is
suspended within another, and is acted on by a couple tending
to turn it about a vertical axis, is probably the best fitted for
absolute measurements. A method of calculating the constants of
such an arrangement is given in Art. 697.
726.] If, however, we wish, by means of a feeble current, to
produce a considerable electromagnetic force, it is better to place
the suspended coil parallel to the fixed coil, and to make it capable
of motion to or from it.
The suspended coil in Dr. Joule's
current- weigher, Fig. 56, is horizontal,
and capable of vertical motion, and the
force between it and the fixed coil is
estimated by the weight which must
be added to or removed from the coil
in order to bring it to the same relative
position with respect to the fixed coil
that it has when no current passes.
The suspended coil may also be
fastened to the extremity of the hori-
56< zontal arm of a torsion-balance, and
may be placed between two fixed coils, one of which attracts it,
while the other repels it, as in Fig. 57.
By arranging the coils as described in Art. 729, the force acting
on the suspended coil may be made nearly uniform within a small
distance of the position of equilibrium.
Another coil may be fixed to the other extremity of the arm
of the torsion-balance and placed between two fixed coils. If the
728.]
CURRENT-WEIGHER.
333
two suspended coils are similar, but with the current flowing in
opposite directions, the effect of terrestrial magnetism on the
Fig. 57.
position of the arm of the torsion-balance will be completely
eliminated.
727.] If the suspended coil is in the shape of a long solenoid,
and is capable of moving parallel to its axis, so as to pass into
the interior of a larger fixed solenoid having the same axis, then,
if the current is in the same direction in both solenoids, the sus
pended solenoid will be sucked into the fixed one by a force which
will be nearly uniform as long as none of the extremities of the
solenoids are near one another.
728.] To produce a uniform longitudinal force on a small coil
placed between two equal coils of much larger dimensions, we
should make the ratio of the diameter of the large coils to the dis
tance between their planes that of 2 to \/3. If we send the same
current through these coils in opposite directions, then, in the ex
pression for o>, the terms involving odd powers of r disappear, and
since sin2 a = -f and cos2 a = f, the term involving /-4 disappears
also, and we have
~ Q2 (0) + V
&c
which indicates a nearly uniform force on a small suspended coil.
The arrangement of the coils in this case is that of the two outer
coils in the galvanometer with three coils, described at Art. 715.
See Fig. 51.
334 ELECTROMAGNETIC INSTRUMENTS. [?29-
729.] If we wish to suspend a coil between two coils placed
so near it that the distance between the mutually acting wires is
small compared with the radius of the coils, the most uniform force
is obtained by making the radius of either of the outer coils exceed
that of the middle one by — - ^ of the distance between the planes
v3
of the middle and outer coils.
CHAPTER XVI.
ELECTROMAGNETIC OBSERVATIONS.
730.] So many of the measurements of electrical quantities
depend on observations of the motion of a vibrating body that we
shall devote some attention to the nature of this motion, and the
best methods of observing it.
The small oscillations of a body about a position of stable equi
librium are, in general, similar to those of a point acted on by
a force varying directly as the distance from a fixed point. In
the case of the vibrating bodies in our experiments there is also
a resistance to the motion, depending on a variety of causes, such
as the viscosity of the air, and that of the suspension fibre. In
many electrical instruments there is another cause of resistance,
namely, the reflex action of currents induced in conducting circuits
placed near vibrating magnets. These currents are induced by the
motion of the magnet, and their action on the magnet is, by the
law of Lenz, invariably opposed to its motion. This is in many
cases the principal part of the resistance.
A metallic circuit, called a Damper, is sometimes placed near
a magnet for the express purpose of damping or deadening its
vibrations. We shall therefore speak of this kind of resistance
as Damping.
In the case of slow vibrations, such as can be easily observed,
the whole resistance, from whatever causes it may arise, appears
to be proportional to the velocity. It is only when the velocity
is much greater than in the ordinary vibrations of electromagnetic
instruments that we have evidence of a resistance proportional to
the square of the velocity.
We have therefore to investigate the motion of a body subject
to an attraction varying as the distance, and to a resistance varying
as the velocity.
336
ELECTROMAGNETIC OBSERVATIONS.
731.] The following application, by Professor Tait*, of the
principle of the Hodograph, enables us to investigate this kind
of motion in a very simple manner by means of the equiangular
spiral.
Let it be required to find the acceleration of a particle which
describes a logarithmic or equiangular spiral with uniform angular
velocity o> about the pole.
The property of this spiral is, that the tangent PT makes with
the radius vector PS a constant angle a.
If v is the velocity at the point P, then
v . sin a = co . SP.
Hence, if we draw SP' parallel to PT and equal to SP, the velocity
at P will be given both in magnitude and direction by
v =
sin a
•SP.
Fig. 58.
Hence P' will be a point in the hodograph. But SP is SP turned
through a constant angle TT — a, so that the hodograph described
by P is the same as the original spiral turned about its pole through
an angle TT — a.
The acceleration of P is represented in magnitude and direction
by the velocity of P' multiplied by the same factor, -.
Hence, if we perform on SP the same operation of turning it
* Proc. R. S. Win., Dec. 16, 1867.
732.] DAMPED VIBRATIONS. 337
through an angle IT — a into the position SP', the acceleration of P
will be equal in magnitude and direction to
-£•'&,
where SP' is equal to SP turned through an angle 2 IT — 2 a.
If we draw PF equal and parallel to SP', the acceleration will be
9
PF, which we may resolve into
sin2 a
J?LpS*n& -4-P*.
sm*a sin^a
The first of these components is a central force towards S pro
portional to the distance.
The second is in a direction opposite to the velocity, and since
_, sin a cos a
PK = 2 cos a PS = - 2 - v,
0}
this force may be written
co cos a
— 2—. v.
sin a
The acceleration of the particle is therefore compounded of two
parts, the first of which is an attractive force /ur, directed towards S,
and proportional to the distance, and the second is — 2 kv, a resist
ance to the motion proportional to the velocity, where
ft)2 , 7 cos a
a = . , and k = o> -.
sin^ a sin a
If in these expressions we make a = — , the orbit becomes a circle,
and we have JUG = o)02, and k = 0.
Hence, if the law of attraction remains the same, ju = /ut0 , and
co = o)0 sin a,
or the angular velocity in different spirals with the same law of
attraction is proportional to the sine of the angle of the spiral.
732.] If we now consider the motion of a point which is the
projection of the moving point P on the horizontal line XT, we
shall find that its distance from S and its velocity are the horizontal
components of those of P. Hence the acceleration of this point is
also an attraction towards S, equal to /x, times its distance from Sf
together with a retardation equal to k times its velocity.
We have therefore a complete construction for the rectilinear
motion of a point, subject to an attraction proportional to the
distance from a fixed point, and to a resistance proportional to
the velocity. The motion of such a point is simply the horizontal
VOL. II. Z
338 ELECTROMAGNETIC OBSERVATIONS. [733.
part of the motion of another point which moves with uniform
angular velocity in a logarithmic spiral.
733.] The equation of the spiral is
r = Ce-$CQia.
To determine the horizontal motion, we put
<£ = co ^, x = a-\-r sin </>,
where a is the value of x for the point of equilibrium.
If we draw BSD making an angle a with the vertical, then the
tangents BX> DY, GZ, &c. will be vertical, and X, Y, Z, &c. will
be the extremities of successive oscillations.
734.] The observations which are made on vibrating bodies are —
(1) The scale-reading at the stationary points. These are called
Elongations.
(2) The time of passing a definite division of the scale in the
positive or negative direction.
(3) The scale-reading at certain definite times. Observations of
this kind are not often made except in the case of vibrations
of long period *.
The quantities which we have to determine are —
(1) The scale-reading at the position of equilibrium.
(2) The logarithmic decrement of the vibrations.
(3) The time of vibration.
To determine the Reading at the Position of Equilibrium from
Three Consecutive Elongations,
735.] Let #!, #2, #3 be the observed scale-readings, corresponding
to the elongations X, Y, Z, and let a be the reading at the position
of equilibrium, S, and let r^ be the value of SB,
# j — a = /! sin a,
$2 — a = — 1\ sin a e~* cot a,
#3 — a = rl sina£-27rcota.
From these values we find
(*!-«) 0*8 -«) = 0*2-«)2»
, X-,
whence a = —
vU\ "J~ «2/o "™ * .— &O
When a*3 does not differ much from x^ we may use as an ap
proximate formula
a = }(a?1 + 2a?a + a?3).
* See Gauss, Resultate des Magnetischen Vereins, 1836. II.
LOGAKITHMIC DECREMENT. 339
To determine the Logarithmic Decrement.
736.] The logarithm of the ratio of the amplitude of a vibration
to that of the next following is called the Logarithmic Decrement.
If we write p for this ratio
L is called the common logarithmic decrement, and A. the Napierian
logarithmic decrement. It is manifest that
A = L loge 10 = 77 cot a.
Hence a = cot"1-
77
which determines the angle of the logarithmic spiral.
In making a special determination of A we allow the body to
perform a considerable number of vibrations. If c1 is the amplitude
of the first, and cn that of the n^ vibration,
If we suppose the accuracy of observation to be the same for
small vibrations as for large ones, then, to obtain the best value
of A, we should allow the vibrations to subside till the ratio of c1 to
cn becomes most nearly equal to e, the base of the Napierian
logarithms. This gives n the nearest whole number to - + 1 .
A
Since, however, in most cases time is valuable, it is best to take
the second set of observations before the diminution of amplitude
has proceeded so far.
737.] In certain cases we may have to determine the position
of equilibrium from two consecutive elongations, the logarithmic
decrement being known from a special experiment. We have then
_ #l + £ ^2
Time of Vibration .
738.] Having determined the scale-reading of the point of equi
librium, a conspicuous mark is placed at that point of the scale,
or as near it as possible, and the times of the passage of this mark
are noted for several successive vibrations.
Let us suppose that the mark is at an unknown but very small
distance as on the positive side of the point of equilibrium, and that
z 2
340 ELECTROMAGNETIC OBSERVATIONS. [739.
tfj is the observed time of the first transit of the mark in the positive
direction, and £2, ^3, &c. the times of the following transits.
If T be the time of vibration, and P15 P2, P3, &c. the times of
transit of the true point of equilibrium,
where vlt v29 &c. are the successive velocities of transit, which we
may suppose uniform for the very small distance SB.
If p is the ratio of the amplitude of a vibration to the next in
succession, 1 , as x
v9 — -- #T , and. — = — p — •
P l /2 ^l
If three transits are observed at times ti3 t2, £3, we find
The period of vibration is therefore
2P+1
The time of the second passage of the true point of equilibrium is
P2 = i (^-f 2 ^2 + O ~i / " \z (*i — 2 ^2 + ^)-
Three transits are sufficient to determine these three quantities,
but any greater number may be combined by the method of least
squares. Thus, for five transits,
The time of the third transit is,
739.] The same method may be extended to a series of any
number of vibrations. If the vibrations are so rapid that the time
of every transit cannot be recorded, we may record the time of
every third or every fifth transit, taking care that the directions
of successive transits are opposite. If the vibrations continue
regular for a long time, we need not observe during the whole
time. We may begin by observing a sufficient number of transits
to determine approximately the period of vibration, T, and the time
of the middle transit, P, noting whether this transit is in the
positive or the negative direction. We may then either go on
counting the vibrations without recording the times of transit,
or we may leave the apparatus un watched. We then observe a
PERIODIC TIME OF VIBRATION. 341
second series of transits,, and deduce the time of vibration T' and
the time of middle transit P', noting the direction of this transit.
If T and Tf, the periods of vibration as deduced from the two
sets of observations, are nearly equal, we may proceed to a more
accurate determination of the period by combining the two series
of observations.
Dividing P'— P by T, the quotient ought to be very nearly
an integer, even or odd according as the transits P and P' are
in the same or in opposite directions. If this is not the case, the
series of observations is worthless, but if the result is very nearly
a whole number n, we divide P'— P by n, and thus find the mean
value of T for the whole time of swinging.
740.] The time of vibration T thus found is the actual mean
time of vibration, and is subject to corrections if we wish to deduce
from it the time of vibration in infinitely small arcs and without
damping.
To reduce the observed time to the time in infinitely small arcs,
we observe that the time of a vibration of amplitude a is in general
of the form T - T^(l + *c2),
where K is a coefficient, which, in the case of the ordinary pendulum,
is -g^. Now the amplitudes of the successive vibrations are c,
cp~1f cp~2, ... cpl~n, so that the whole time of n vibrations is
where T is the time deduced from the observations.
Hence, to find the time T^ in infinitely small arcs, we have
approximately,
n p-!
To find the time T0 when there is no damping, we have
sn a
741.] The equation of the rectilinear motion of a body, attracted
to a fixed point and resisted by a force varying as the velocity, is
7 n j
.^ + 2*^+»*(*-«)=sO, (1)
where x is the coordinate of the body at the time t, and a is the
coordinate of the point of equilibrium.
342 ELECTROMAGNETIC OBSERVATIONS. [?42-
To solve this equation, let
x-a = e-Vy; (2)
then gl + ^.^^o; (3)
the solution of which is
y — Ccos (\/oo2— IP t-\-d), when k is less than <o ; (4)
y = A + Bt, when k is equal to o> ; (5)
and y — C' cos h ( Vk* — o>2 1 + a), when k is greater than o>. (6)
The value of a? may be obtained from that of y by equation (2).
When k is less than o>, the motion consists of an infinite series of
oscillations, of constant periodic time, but of continually decreasing
amplitude. As k increases, the periodic time becomes longer, and
the diminution of amplitude becomes more rapid.
When k (half the coefficient of resistance) becomes equal to or
greater than o>, (the square root of the acceleration at unit distance
from the point of equilibrium,) the motion ceases to be oscillatory,
and during the whole motion the body can only once pass through
the point of equilibrium, after which it reaches a position of greatest
elongation, and then returns towards the point of equilibrium, con
tinually approaching, but never reaching it.
Galvanometers in which the resistance is so great that the motion
is of this kind are called dead beat galvanometers. They are useful
in many experiments, but especially in telegraphic signalling, in
which the existence of free vibrations would quite disguise the
movements which are meant to be observed.
Whatever be the values of k and o>, the value of a, the scale-
reading at the point of equilibrium, may be deduced from five scale-
readings, p, q, r, s, t, taken at equal intervals of time, by the formula
(p-2+r) (r- 2s + 1) - (q-
On the Observation of the Galvanometer.
742.] To measure a constant current with the tangent galvano
meter, the instrument is adjusted with the plane of its coils parallel
to the magnetic meridian, and the zero reading is taken. The
current is then made to pass through the coils, and the deflexion
of the magnet corresponding to its new position of equilibrium is
observed. Let this be denoted by $.
Then, if // is the horizontal magnetic force, G the coefficient of
the galvanometer, and y the strength of the current,
(I)
744-] DEFLEXION OF THE GALVANOMETER. 343
If the coefficient of torsion of the suspension fibre is r MH (see
Art. 452), we must use the corrected formula
JT
y = -(tan$+r(j[>sec<£). (2)
Best Value of the Deflexion.
743.] In some galvanometers the number of windings of the
coil through which the current flows can be altered at pleasure.
In others a known fraction of the current can be diverted from the
galvanometer by a conductor called a Shunt. In either case the
value of G, the effect of a unit-current on the magnet, is made
to vary.
Let us determine the value of £, for which a given error in the
observation of the deflexion corresponds to the smallest error of the
deduced value of the strength of the current.
Differentiating equation (1), we find
dy H , .
4=^sec*-
Eliminating G, -~ = — sin 2 $. (4)
This is a maximum for a given value of y when the deflexion is
45°. The value of G should therefore be adjusted till Gy is as
nearly equal to H as is possible ; so that for strong currents it is
better not to use too sensitive a galvanometer.
On the Best Method of applying the Current.
744.] When the observer is able, by means of a key, to make or
break the connexions of the circuit at any instant, it is advisable to
operate with the key in such a way as to make the magnet arrive
at its position of equilibrium with the least possible velocity. The
following method was devised by Gauss for this purpose.
Suppose that the magnet is in its position of equilibrium, and that
there is no current. The observer now makes contact for a short
time, so that the magnet is set in motion towards its new position
of equilibrium. He then breaks contact. The force is now towards
the original position of equilibrium, and the motion is retarded. If
this is so managed that the magnet comes to rest exactly at the
new position of equilibrium,, and if the observer again makes con
tact at that instant and maintains the contact, the magnet will
remain at rest in its new position.
344 ELECTROMAGNETIC OBSERVATIONS. [745.
If we neglect the effect of the resistances and also the inequality
of the total force acting in the new and the old positions, then,
since we wish the new force to generate as much kinetic energy
during the time of its first action as the original force destroys
while the circuit is broken, we must prolong the first action of the
current till the magnet has moved over half the distance from the
first position to the second. Then if the original force acts while
the magnet moves over the other half of its course, it will exactly
stop it. Now the time required to pass from a point of greatest
elongation to a point half way to the position of equilibrium is
one-sixth of a complete period, or one-third of a single vibration.
The operator, therefore, having previously ascertained the time
of a single vibration, makes contact for one-third of that time,
breaks contact for another third of the same time, and then makes
contact again during the continuance of the experiment. The
magnet is then either at rest, or its vibrations are so small that
observations may be taken at once, without waiting for the motion
to die away. For this purpose a metronome may be adjusted so as
to beat three times for each single vibration of the magnet.
The rule is somewhat more complicated when the resistance is of
sufficient magnitude to be taken into account, but in this case the
vibrations die away so fast that it is unnecessary to apply any
corrections to the rule.
When the magnet is to be restored to its original position, the
circuit is broken for one-third of a vibration, made again for an
equal time, and finally broken. This leaves the magnet at rest in
its former position.
If the reversed reading is to be taken immediately after the direct
one, the circuit is broken for the time of a single vibration and
then reversed. This brings the magnet to rest in the reversed
position.
Measurement l>y the First Swing.
745.] When there is no time to make more than one observation,
the current may be measured by the extreme elongation observed
in the first swing of the magnet. If there is no resistance, the
permanent deflexion $ is half the extreme elongation. If the re
sistance is such that the ratio of one vibration to the next is p, and
if 00 is the zero reading, and dl the extreme elongation in the first
swing, the deflexion, <£, corresponding to the point of equilibrium is
0Q+P0!
9 1+p
747-] SERIES OF OBSERVATION'S. 345
In this way the deflexion may be calculated without waiting for
the magnet to come to rest in its position of equilibrium.
To make a Series of Observations.
746.] The best way of making a considerable number of mea
sures of a constant current is by observing three elongations while
the current is in the positive direction, then breaking contact for
about the time of a single vibration, so as to let the magnet swing
into the position of negative deflexion, then reversing the current
and observing three successive elongations on the negative side,
then breaking contact for the time of a single vibration and re
peating the observations on the positive side, and so on till a suffi
cient number of observations have been obtained. In this way the
errors which may arise from a change in the direction of the earth's
magnetic force during the time of observation are eliminated. The
operator, by carefully timing the making and breaking of contact,
can easily regulate the extent of the vibrations, so as to make them
sufficiently small without being indistinct. The motion of the
magnet is graphically represented in Fig. 59, where the abscissa
represents the time, and the ordinate the deflexion of the magnet.
If 01 . . . 06 be the observed elongations, the deflexion is given by the
equation 8 = + 2 0 + 0_0_20 — 0.
Fig. 59.
Method of Multiplication.
747.] In certain cases, in which the deflexion of the galvanometer
magnet is very small, it may be advisable to increase the visible
effect by reversing the current at proper intervals, so as to set
up a swinging motion of the magnet. For this purpose, after
ascertaining the time, T, of a single vibration of the magnet, the
current is sent in the positive direction for a time T, then in the
reversed direction for an equal time, and so on. When the motion
of the magnet has become visible, we may make the reversal of the
current at the observed times of greatest elongation.
Let the magnet be at the positive elongation 00, and let the
current be sent through the coil in the negative direction. The
346 ELECTROMAGNETIC OBSERVATIONS. [748.
point of equilibrium is then — $, and the magnet will swing to a
negative elongation 0, such that
Similarly, if the current is now made positive while the magnet
swings to 02, P02 = -01 + (p+ 1) 0,
or P202 = 00 + (P+1)24>;
and if the current is reversed n times in succession, we find
whence we may find <£ in the form
**«*FTf=7*-
If ^ is a number so great that p~n may be neglected, the ex
pression becomes n — 1
The application of this method to exact measurement requires an
accurate knowledge of p, the ratio of one vibration of the magnet
to the next under the influence of the resistances which it expe
riences. The uncertainties arising from the difficulty of avoiding
irregularities in the value of p generally outweigh the advantages
of the large angular elongation. It is only where we wish to
establish the existence of a very small current by causing it to
produce a visible movement of the needle that this method is really
valuable.
On the Measurement of Transient Currents.
748.] When a current lasts only during a very small fraction of
the time of vibration of the galvanometer-magnet, the whole quan
tity of electricity transmitted by the current may be measured by
the angular velocity communicated to the magnet during the
passage of the current, and this may be determined from the
elongation of the first vibration of the magnet.
If we neglect the resistance which damps the vibrations of the
magnet, the investigation becomes very simple.
Let y be the intensity of the current at any instant, and Q the
quantity of electricity which it transmits, then
= \ydt. (1)
749-] TRANSIENT CURRENTS. 347
Let M be the magnetic moment, and A the moment of inertia of
the magnet and suspended apparatus,
,72/9
A "L^ + MHsm 0 = MGy cos 0. (2)
(It
If the time of the passage of the current is very small, we may
integrate with respect to t during this short time without regarding
the change of 0, and we find
=MG cos 00 ydt + C = MGQ cos 00 + C. (3)
This shews that the passage of the quantity Q produces an angular
momentum MGQ cos 00 in the magnet, where 00 is the value of 0
at the instant of passage of the current. If the magnet is initially
in equilibrium, we may make 00 = 0.
The magnet then swings freely and reaches an elongation 01. If
there is no resistance, the work done against the magnetic force
during this swing is MR (I — cosflj.
The energy communicated to the magnet by the current is
Equating these quantities, we find
lf = 2^(l-cos<y, (4)
s-IJ- a ^ •*• » '
tf6- -^t
dO /MH .
whence -=- = 2 A / — — - sin J 0j
^ \ A
i\/rn
Qby(3). (5)
A
But if T be the time of a single vibration of the magnet,
T
= " A/
(6)
TT m
and we find Q = — - - 2 sin \ Qlt (7)
where // is the horizontal magnetic force, Q- the coefficient of the
galvanometer, T the time of a single vibration, and Ol the first-
elongation of the magnet.
749.] In many actual experiments the elongation is a small
angle, and it is then easy to take into account the effect of resist
ance, for we may treat the equation of motion as a linear equation.
Let the magnet be at rest at its position of equilibrium, let an
angular velocity v be communicated to it instantaneously, and let
its first elongation be Ol .
348 ELECTROMAGNETIC OBSERVATIONS. [750.
The equation of motion is
(8)
— = C^secpe-^t^Pcosfa t + p). (9)
cl-t
,j a
When t = 0, 6 = 0, and — = C(dl = v.
dt
When <*>!$ + p = ->
Hence 0, = -- e v* cos/3. (11)
ME
JNow — — = or = o>i sec^/3, (12)
^4
x
tan£ = -, wj^^, (13)
7T jt-i
Hence ' *1 = , (l.)
-
which gives the first elongation in terms of the quantity of elec
tricity in the transient current, and conversely, where T^ is the
observed time of a single vibration as affected by the actual resist
ance of damping. When A. is small we may use the approximate
formula TT T
Method of Recoil.
750.] The method given above supposes the magnet to be at
rest in its position of equilibrium when the transient current is
passed through the coil. If we wish to repeat the experiment
we must wait till the magnet is again at rest. In certain cases,
however, in which we are able to produce transient currents of
equal intensity, and to do so at any desired instant, the following
method, described by Weber *, is the most convenient for making
a continued series of observations.
* Rcsullate des Magnetisckcn Vereins, 1838, p. 98.
75O.] METHOD OF KECOIL. 349
Suppose that we set the magnet swinging by means of a transient
current whose value is QQ. If, for brevity, we write
G V^TT2 -itan-i£
Jf~—T~~e n = jSr' (18)
then the first elongation
^ = KQ, = ^ (say). (19)
The velocity instantaneously communicated to the magnet at
starting is jf Q
v-'^rft- (20)
When it returns through the point of equilibrium in a negative
direction its velocity will be
v1 =—ve~^. (21)
The next negative elongation will be
6z = -61e-* = b1. (22)
When the magnet returns to the point of equilibrium, its velocity
will be V2 = V()e-2\ (23)
Now let an instantaneous current, whose total quantity is — Q,
be transmitted through the coil at the instant when the magnet is
at the zero point. It will change the velocity v2 into v2— v, where
If Q is greater than Q0e~2^, the new velocity will be negative and
equal to
^^ VH5 "BO*
The motion of the magnet will thus be reversed, and the next
elongation will be negative,
03 = — K(Q — Q06~2A) = c1= —KQ + O^^. (25)
The magnet is then allowed to come to its positive elongation
and when it again reaches the point of equilibrium a positive
current whose quantity is Q is transmitted. This throws the
magnet back in the positive direction to the positive elongation
or, calling this the first elongation of a second series of four,
#2 = KQ (1 — <?~"2A)-f a^e~^K. (28)
Proceeding in this way, by observing two elongations + and — ,
then sending a positive current and observing two elongations
350
ELECTROMAGNETIC OBSERVATIONS.
[751-
— and -f , then sending a positive current, and so on, we obtain
a series consisting of sets of four elongations, in each of which
and
(29)
(30)
If n series of elongations have been observed, then we find the
logarithmic decrement from the equation
and Q from the equation
. (32)
Fig, 60.
The motion of the magnet in the method of recoil is graphically
represented in Fig. 60, where the abscissa represents the time, and
the ordinate the deflexion of the magnet at that time. See Art. 760.
Method of Multiplication.
751.] If we make the transient current pass every time that the
magnet passes through the zero point, and always so as to increase
the velocity of the magnet, then, if 01} 02, &c. are the successive
elongations, ^ = -KQ-e~* Olf (33)
Os=-KQ-e-^e2. (34)
The ultimate value to which the elongation tends after a great
many vibrations is found by putting 0n = — Qn-i > whence we find
(35)
If A is small, the value of the ultimate elongation may be large,
but since this involves a long continued experiment, and a careful
determination of A, and since a small error in A introduces a large
error in the determination of Q, this method is rarely useful for
75I-] MISTIMING THE CURRENT. 351
numerical determination, and should be reserved for obtaining- evi
dence of the existence or non-existence of currents too small to be
observed directly.
In all experiments in which transient currents are made to act on
the moving1 magnet of the galvanometer, it is essential that the
whole current should pass while the distance of the magnet from
the zero point remains a small fraction of the total elongation.
The time of vibration should therefore be large compared with the
time required to produce the current, and the operator should have
his eye on the motion of the magnet, so as to regulate the instant
of passage of the current by the instant of passage of the magnet
through its point of equilibrium.
To estimate the error introduced by a failure of the operator to
produce the current at the proper instant, we observe that the effect
of a force in increasing the elongation varies as
and that this is a maximum when 0 = 0. Hence the error arising
from a mistiming of the current will always lead to an under
estimation of its value, and the amount of the error may be
estimated by comparing the cosine of the phase of the vibration at
the time of the passage of the current with unity.
CHAPTER XVII.
COMPARISON OF COILS.
Experimental Determination of the Electrical Constants
of a Coil.
752.] WE have seen in Art. 717 that in a sensitive galvanometer
the coils should he of small radius, and should contain many
windings of the wire. It would he extremely difficult to determine
the electrical constants of such a coil hy direct measurement of its
form and dimensions, even if we could obtain access to every
winding of the wire in order to measure it. But in fact the
greater number of the windings are not only completely hidden
by the outer windings, but we are uncertain whether the pressure
of the outer windings may not have altered the form of the inner
ones after the coiling of the wire.
It is better therefore to determine the electrical constants of the
coil by direct electrical comparison with a standard coil whose con
stants are known.
Since the dimensions of the standard coil must be determined by
actual measurement, it must be made of considerable size, so that
the unavoidable error of measurement of its diameter or circum
ference may be as small as possible compared with the quantity
measured. The channel in which the coil is wound should be of
rectangular section, and the dimensions of the section should be
small compared with the radius of the coil. This is necessary, not
so much in order to diminish the correction for the size of the
section, as to prevent any uncertainty about the position of those
windings of the coil which are hidden by the external windings *.
* Large tangent galvanometers are sometimes made with a single circular con
ducting ring of considerable thickness, which is sufficiently stiff to maintain its form
without any support. This is not a good plan for a standard instrument. The dis
tribution of the current within the conductor depends on the relative conductivity
753-] PRINCIPAL CONSTANTS OF A COIL. 353
The principal constants which we wish to determine are —
(1) The magnetic force at the centre of the coil due to a unit-
current. This is the quantity denoted by G1 in Art. 700.
(2) The magnetic moment of the coil due to a unit-current.
This is the quantity ff1 .
753.] To determine G1. Since the coils of the working galva
nometer are much smaller than the standard coil, we place the
galvanometer within the standard coil, so that their centres coincide,
the planes of both coils being vertical and parallel to the earth's
magnetic force. We have thus obtained a differential galvanometer
one of whose coils is the standard coil, for which the value of G±
is known, while that of the other coil is £/, the value of which we
have to determine.
The magnet suspended in the centre of the galvanometer coil
is acted on by the currents in both coils. If the strength of the
current in the standard coil is y, and that in the galvanometer coil
y', then, if these currents flowing in opposite directions produce a
deflexion 6 of the magnet,
#tan8= G^y'-Gl7, (1)
where H is the horizontal magnetic force of the earth.
If the currents are so arranged as to produce no deflexion, we
may find <?/ by the equation
<?/= -, e,. (2)
We may determine the ratio of y to y in several ways. Since the
value of Gl is in general greater for the galvanometer than for the
standard coil, we may arrange the circuit so that the whole current
y flows through the standard coil, and is then divided so that y'
flows through the galvanometer and resistance coils, the combined
resistance of which is J?13 while the remainder y — y' flows through
another set of resistance coils whose combined resistance is E .
of its various parts. Hence any concealed flaw in the continuity of the metal may
cause the main stream of electricity to flow either close to the outside or close to the
inside of the circular ring. Thus the true path of the current becomes uncertain.
Besides this, when the current flows only once round the circle, especial care is
necessary to avoid any action on the suspended magnet due to the current on its
way to or from the circle, because the current in the electrodes is equal to that in
the circle. In the construction of many instruments the action of this part of the
current seems to have been altogether lost sight of.
The most perfect method is to make one of the electrodes in the form of a metal
tube, and the other a wire covered with insulating material, and placed inside the
tube and concentric with it. The external action of the electrodes when thus arranged
is zero, by Art. 683.
VOL. II. A a
354 COMPARISON OF COILS. [754-
We have then, by Art. 276,
or = . (4)
V H-i
and G;=^+^Gl. (5)
tf2
If there is any uncertainty about the actual resistance of the
galvanometer coil (on account, say, of an uncertainty as to its tem
perature) we may add resistance coils to it, so that the resistance of
the galvanometer itself forms but a small part of Hlt and thus
introduces but little uncertainty into the final result.
754.] To determine glt the magnetic moment of a small coil due
to a unit-current flowing through it, the magnet is still suspended
at the centre of the standard coil, but the small coil is moved
parallel to itself along the common axis of both coils, till the same
current, flowing in opposite directions round the coils, no longer
deflects the magnet. If the distance between the centres of the
coils is r, we have now
£ =24 + 3^+4^f +&c. (6)
^.O ^.4 £>O
By repeating the experiment with the small coil on the opposite
side of the standard coil, and measuring the distance between the
positions of the small coil, we eliminate the uncertain error in the
determination of the position of the centres of the magnet and
of the small coil, and we get rid of the terms in g2) g±, &c.
If the standard coil is so arranged that we can send the current
through half the number of windings, so as to give a different value
to G19 we may determine a new value of r, and thus, as in Art. 454,
we may eliminate the term involving g^ .
It is often possible, however, to determine gz by direct measure
ment of the small coil with sufficient accuracy to make it available
in calculating the value of the correction to be applied to g^ in
the equation i
where #3 = — -ir0a(6«2-f 3f2 — 2»j2), by Art. 700.
o
Comparison of Coefficients of Induction.
755.] It is only in a small number of cases that the direct
calculation of the coefficients of induction from the form and
755-]
MUTUAL INDUCTION OF TWO COILS.
355
position of the circuits can be easily performed. In order to attain
a sufficient degree of accuracy, it is necessary that the distance
between the circuits should be capable of exact measurement.
But when the distance between the circuits is sufficient to prevent
errors of measurement from introducing large errors into the result,
the coefficient of induction itself is necessarily very much reduced
in magnitude. Now for many experiments it is necessary to make
the coefficient of induction large, and we can only do so by bringing
the circuits close together, so that the method of direct measure
ment becomes impossible, and, in order to determine the coefficient
of induction, we must compare it with that of a pair of coils ar
ranged so that their coefficient may be obtained by direct measure
ment and calculation.
This may be done as follows :
Let A and a be the standard
pair of coils, B and b the coils to
be compared with them. Con
nect A and B in one circuit, and
place the electrodes of the gal
vanometer, G, at P and Q, so
that the resistance of PAQ is
R, and that of QBP is S, K
being the resistance of the gal
vanometer. Connect a and b in
one circuit with the battery. Fig. 51.
Let the current in A be »,
that in B, y> and that in the galvanometer, sc —y, that in the battery
circuit being y.
Then, if Ml is the coefficient of induction between A and «, and
M2 that between B and b, the integral induction current through
the galvanometer at breaking the battery circuit is
x-y - y
R" S
1 +
(8)
.
R "" 8
By adjusting the resistances R and 8 till there is no current
through the galvanometer at making or breaking the galvanometer
circuit, the ratio of M2 to M1 may be determined by measuring that
of S to R.
A a 2
356 COMPARISON OF COILS. [756.
Comparison of a Coefficient of Self-induction with a Coefficient of
Mu tual Induction .
756.] In the branch AF of Wheatstone's Bridge let a coil be
inserted, the coefficient of self-induc
tion of which we wish to find. Let
us call it L.
In the connecting wire between A
and the battery another coil is inserted.
The coefficient of mutual induction be
tween this coil and the coil in AF
is M. It may be measured by the
method described in Art. 755.
If the current from A to F is #, and
.p. 62 that from A to H is ^, that from Z
to A, through B, will be oc+y. The
external electromotive force from A to F is
The external electromotive force along AH is
A-H=Qy. (10)
If the galvanometer placed between F and H indicates no current,
either transient or permanent, then by (9) and (10), since I1 — F=0,
whence L = - (l + ~) M. (13)
^o
Since L is always positive, M must be negative, and therefore the
current must flow in opposite directions through the coils placed
in P and in B. In making the experiment we may either begin
by adjusting the resistances so that
PS=QR, (14)
which is the condition that there may be no permanent current,
and then adjust the distance between the coils till the galvanometer
ceases to indicate a transient current on making and breaking the
battery connexion ; or, if this distance is not capable of adjustment,
we may get rid of the transient current by altering the resistances
Q and S in such a way that the ratio of Q to S remains constant.
If this double adjustment is found too troublesome, we may adopt
757-] SELF-INDUCTION. 357
a third method. Beginning with an arrangement in which the
transient current due to self-induction is slightly in excess of that
due to mutual induction, we may get rid of the inequality by in
serting a conductor whose resistance is W between A and Z. The
condition of no permanent current through the galvanometer is not
affected by the introduction of W. We may therefore get rid of
the transient current by adjusting the resistance of W alone. When
this is done the value of L is
. (15)
.
Comparison of the Coefficients of Self -induction of Two Coils.
757.] Insert the coils in two adjacent branches of Wheatstone's
Bridge. Let L and N be the coefficients of self-induction of the
coils inserted in P and in R respectively, then the condition of no
galvanometer current is
(P* + l^)8y=Qy(X* + N%), (16)
whence PS = QJR, for no permanent current, (17)
and — = — , for no transient current. (18)
JT J-l/
Hence, by a proper adjustment of the resistances, both the per
manent and the transient current can be got rid of, and then
the ratio of L to N can be determined by a comparison of the
resistances.
CHAPTER XVIIL
ELECTROMAGNETIC UNIT OF RESISTANCE.
On the Determination of the Resistance of a Coil in Electro
nic Measure.
758.] THE resistance of a conductor is defined as the ratio of the
numerical value of the electromotive force to that of the current
which it produces in the conductor. The determination of the
value of the current in electromagnetic measure can be made by
means of a standard galvanometer, when we know the value of the
earth's magnetic force. The determination of the value of the
electromotive force is more difficult, as the only case in which we
can directly calculate its value is when it arises from the relative
motion of the circuit with respect to a known magnetic system.
759.] The first determination of the resistance of a wire in
electromagnetic measure was made by Kirchhoff*. He employed
two coils of known form, A1 and A^ and calculated their coefficient
of mutual induction from the geo
metrical data of their form and
position. These coils were placed
in circuit with a galvanometer, 6r,
and a battery, B, and two points
of the circuit, P, between the coils,
and Q, between the battery and
galvanometer, were joined by the
wire whose resistance, R, was to be measured.
When the current is steady it is divided between the wire and
the galvanometer circuit, and produces a certain permanent de
flexion of the galvanometer. If the coil A1 is now removed quickly
* * Bestimmong Her Constanten von welcher die Intensitat inducirter elektrischer
Strome abhangt.' Pogg. Ann., Ixxvi (April 1849).
759-] KIRCHHOFF'S METHOD. 359
from A2 and placed in a position in which the coefficient of mutual
induction between Al and A.2 is zero (Art. 538), a current of induc
tion is produced in both circuits, and the galvanometer needle
receives an impulse which produces a certain transient deflexion.
The resistance of the wire, R, is deduced from a comparison
between the permanent deflexion, due to the steady current, and the
transient deflexion, due to the current of induction.
Let the resistance of QGAl P be K, of PA2 £Q, B, and of PQ, R.
Let Lj M and N be the coefficients of induction of Al and A2.
Let x be the current in (7, and y that in J3, then the current
from P to Q is x— y.
Let E be the electromotive force of the battery, then
)= o, (l)
Rx + (B + R}y + -j- (Mx + Ny} = E. (2)
When the currents are constant, and everything at rest,
(K+R}x-Ry = 0. (3)
If M now suddenly becomes zero on account of the separation of
A1 from A2 , then, integrating with respect to t,
J «/ "" \ /
— Mx = lEdt = 0. (5)
whence x = M(B\.R]^ ml ^2 ' (6)
Substituting the value of y in terms of x from (3), we find
6 = ~R (B + R)(K+R}-R? (7)
When, as in Kirchhoff 's experiment, both B and K are large
compared with R, this equation is reduced to
x _M
~x~~R'
Of these quantities, x is found from the throw of the galvanometer
due to the induction current. See Art. 768. The permanent cur
rent, at, is found from the permanent deflexion due to the steady
current; see Art. 746. M is found either by direct calculation
from the geometrical data, or by a comparison with a pair of coils,
for which this calculation has been made; see Art. 755. From
360 UNIT OF RESISTANCE. [760.
these three quantities R can be determined in electromagnetic mea
sure.
These methods involve the determination of the period of vibra
tion of the galvanometer magnet, and of the logarithmic decrement
of its oscillations.
Weber's Method by Transient Currents*.
760.] A coil of considerable size is mounted on an axle, so as to
be capable of revolving about a vertical diameter. The wire of this
coil is connected with that of a tangent galvanometer so as to form
a single circuit. Let the resistance of this circuit be R. Let the
large coil be placed with its positive face perpendicular to the
magnetic meridian, and let it be quickly turned round half a revo
lution. There will be an induced current due to the earth's mag
netic force, and the total quantity of electricity in this current in
electromagnetic measure will be
where ffl is the magnetic moment of the coil for unit current, which
in the case of a large coil may be determined directly, by mea
suring the dimensions of the coil, and calculating the sum of the
areas of its windings. If is the horizontal component of terrestrial
magnetism, and R is the resistance of the circuit formed by the
coil and galvanometer together. This current sets the magnet of
the galvanometer in motion.
If the magnet is originally at rest, and if the motion of the coil
occupies but a small fraction of the time of a vibration of the
magnet, then, if we neglect the resistance to the motion of the
magnet, we have, by Art. 748,
// T
<2=^-2sinU (2)
Cr 7T
where G is the constant of the galvanometer, T is the time of
vibration of the magnet, and 6 is the observed elongation. From
these equations we obtain
* = *°'¥15&' . (3)
The value of H does not appear in this result, provided it is the
same at the position of the coil and at that of the galvanometer.
This should not be assumed to be the case, but should be tested by
comparing the time of vibration of the same magnet, first at one of
these places and then at the other.
* ElcU. Moots*. ; or Pogg., Ann. Ixxxii, 337 (1851).
762.] WEBER'S METHOD. 361
761.] To make a series of observations Weber began with the
coil parallel to the magnetic meridian. He then turned it with its
positive face north, and observed the first elongation due to the
negative current. He then observed the second elongation of the
freely swinging magnet, and on the return of the magnet through
the point of equilibrium he turned the coil with its positive face
south. This caused the magnet to recoil to the positive side. The
series Was continued as in Art. 750, and the result corrected for
resistance. In this way the value of the resistance of the combined
circuit of the coil and galvanometer was ascertained.
In all such experiments it is necessary, in order to obtain suffi
ciently large deflexions, to make the wire of copper, a metal which,
though it is the best conductor, has the disadvantage of altering
considerably in resistance with alterations of temperature. It is
also very difficult to ascertain the temperature of every part of the
apparatus. Hence, in order to obtain a result of permanent value
from such an experiment, the resistance of the experimental circuit
should be compared with that of a carefully constructed resistance-
coil, both before and after each experiment.
Weber's Method by observing the Decrement of the Oscillations
of a Magnet.
762.] A magnet of considerable magnetic moment is suspended
at the centre of a galvanometer coil. The period of vibration and
the logarithmic decrement of the oscillations is observed, first with
the circuit of the galvanometer open, and then with the circuit
closed, and the conductivity of the galvanometer coil is deduced
from the effect which the currents induced in it by the motion of
the magnet have in resisting that motion.
If T is the observed time of a single vibration, and A. the Na
pierian logarithmic decrement for each single vibration, then, if we
write ,,
o> = ^> (1)
and a = ~ , (2)
the equation of motion of the magnet is of the form
$ = Ce-atcos(o>t + (3}. (3)
This expresses the nature of the motion as determined by observa
tion. We must compare this with the dynamical equation of
motion.
362 UNIT OF RESISTANCE. [?62.
Let M be the coefficient of induction between the galvanometer
coil and the suspended magnet. It is of the form
M = Giffi Qi TO + $222 $2 W + &c., (4)
where G1} G2, &c. are coefficients belonging to the coil, ffl3gz, &c.
to the magnet, and Ql(0), Q.2(Q), &c., are zonal harmonics. of the
angle between the axes of the coil and the magnet. See Art. 700.
By a proper arrangement of the coils of the galvanometer, and by
building up the suspended magnet of several magnets placed side by
side at proper distances, we may cause all the terms of M after the
first to become insensible compared with the first. If we also put
(f> = -- 0, we may write
M = Gm sin$, (5)
where G is the principal coefficient of the galvanometer, m is the
magnetic moment of the magnet, and $ is the angle between the
axis of the magnet and the plane of the coil, which, in this ex
periment, is always a small angle.
If I/ is the coefficient of self-induction of the coil, and R its
resistance, and y the current in the coil,
0, (6)
or L~ -fj^y-f £mcos(£ - = 0. (7)
U/t Cit
The moment of the force with which the current y acts on the
magnet is y —r— , or Gmy cos $. The angle </> is in this experiment
ct cp
so small, that we may suppose cos <£ = 1 .
Let us suppose that the equation of motion of the magnet when
the circuit is broken is
where A is the moment of inertia of the suspended apparatus, S~-
Cvv
expresses the resistance arising from the viscosity of the air and
of the suspension fibre, &c., and C<$> expresses the moment of the
force arising from the earth's magnetism, the torsion of the sus
pension apparatus, &c., tending to bring the magnet to its position
of equilibrium.
The equation of motion, as affected by the current, will be
A + sc
762.] WEBER'S METHOD. 363
To determine the motion of the magnet, we have to combine this
equation with (7) and eliminate y. The result is
a linear differential equation of the third order.
We have no occasion, however, to solve this equation, because
the data of the problem are the observed elements of the motion
of the magnet, and from these we have to determine the value
of E.
Let a0 and o)0 be the values of a and o> in equation (2) when the
circuit is broken. In this case R is infinite, and the equation is
reduced to the form (8). We thus find
B=2AaQ, C=A(a^ + ^). (11)
Solving equation (10) for R, and writing
we find
— o), where i=V — I, (12)
Since the value of co is in general much greater than that of a,
the best value of R is found by equating the terms in i o>,
0
2A(a — a0) a-a0 '
We may also obtain a value of R by equating the terms not
involving i, but as these terms are small, the equation is useful
only as a means of testing the accuracy of the observations. From
these equations we find the following testing equation,
(co2-o>02)2}. (15)
Since LAv? is very small compared with G2m2, this equation
a02-a2; (16)
and equation (14) may be written
E=GV_ L
2A(a-a0) r
In this expression G may be determined either from the linear
measurement of the galvanometer coil, or better, by comparison
with a standard coil, according to the method of Art. 753. A is
the moment of inertia of the magnet and its suspended apparatus,
which is to be found by the proper dynamical method. o>, &>0, a
and a0, are given by observation.
364 UNIT OF RESISTANCE. [763.
The determination of the value of m, the magnetic moment of
the suspended magnet, is the most difficult part of the investigation,
because it is affected by temperature, by the earth's magnetic force,
and by mechanical violence, so that great care must be taken to
measure this quantity when the magnet is in the very same circum
stances as when it is vibrating.
The second term of R, that which involves L, is of less import
ance, as it is generally small compared with the first term. The
value of L may be determined either by calculation from the known
form of the coil, or by an experiment on the extra-current of in
duction. See Art. 756.
Thomson's Method by a Revolving Coil.
763.] This method was suggested by Thomson to the Committee
of the British Association on Electrical Standards, and the ex
periment was made by M. M. Balfour Stewart, Fleeming Jenkin,
and the author in 1863 *.
A circular coil is made to revolve with uniform velocity about a
vertical axis. A small magnet is suspended by a silk fibre at the
centre of the coil. An electric current is induced in the coil by
the earth's magnetism, and also by the suspended magnet. This
current is periodic, flowing in opposite directions through the wire
of the coil during different parts of each revolution, but the effect of
the current on the suspended magnet is to produce a deflexion from
the magnetic meridian in the direction of the rotation of the coil.
764.] Let H be the horizontal component of the earth's mag
netism.
Let y be the strength of the current in the coil.
g the total area inclosed by all the windings of the wire.
G the magnetic force at the centre of the coil due to unit-
current.
L the coefficient of self-induction of the coil.
M the magnetic moment of the suspended magnet.
0 the angle between the plane of the coil and the magnetic
meridian.
</> the angle between the axis of the suspended magnet and
the magnetic meridian
A the moment of inertia of the suspended magnet.
MHr the coefficient of torsion of the suspension fibre,
a the azimuth of the magnet when there is no torsion.
R the resistance of the coil.
* See Report of (he British Association for 1863.
765.] THOMSON'S METHOD. 365
The kinetic energy of the system is
T=\Ly* -Hgy sm6-MGy sin (0— <f>) + MHcoaQ+b Atf>. (1 )
The first term, Jrj&y2, expresses the energy of the current as
depending on the coil itself. The second term depends on the
mutual action of the current and terrestrial magnetism, the third
on that of the current and the magnetism of the suspended magnet,
the fourth on that of the magnetism of the suspended magnet and
terrestrial magnetism, and the last expresses the kinetic energy of
the matter composing the magnet and the suspended apparatus
which moves with it.
The potential energy of the suspended apparatus arising from the
torsion of the fibre is
**-S*0. (2)
The electromagnetic momentum of the current is
clT
(6-<t)), (3)
dy
and if R is the resistance of the coil, the equation of the current is
or, since 6 = tot, (5)
<p)cos(0—(})). (6)
765.] It is the result .alike of theory and observation that <£, the
azimuth of the magnet, is subject to two kinds of periodic variations.
One of these is a free oscillation, whose periodic time depends on
the intensity of terrestrial magnetism, and is, in the experiment,
several seconds. The other is a forced vibration whose period is
half that of the revolving coil, and whose amplitude is, as we shall
see, insensible. Hence, in determining y, we may treat $ as
sensibly constant.
We thus find
y = j/^tftf (Hcos6 + La> sin 0) (7)
£(« (8)
+ Ce'*'. (9)
The last term of this expression soon dies away when the rota
tion is continued uniform.
366 UNIT OF RESISTANCE. [766.
The equation of motion of the suspended magnet is
d*T _dT_ f!F_0
d<j> dt dfy dcf)
whence A$— MGy cos (0 — c/>)-f Jf # (sin c/> + r (c/> — a)) = 0. (11)
Substituting the value of y, and arranging the terms according
to the functions of multiples of 6, then we know from observation
that
<£ r= c/>0-f be~lt cos nt + c cos 2 (0— /3), (12)
where c/>0 is the mean value of c/>, and the second term expresses
the free vibrations gradually decaying, and the third the forced
vibrations arising from the variation of the deflecting current.
TT~\T
The value of n in equation (12) is — j- secc/>. That of c, the am-
A.
n2
plitude of the forced vibrations, is J —3- sin c/>. Hence, when the
co
coil makes many revolutions during one free vibration of the magnet,
the amplitude of the forced vibrations of the magnet is very small,
and we may neglect the terms in (11) which involve c.
Beginning with the terms in (11) which do not involve 0, we find
MHGgu /z> J v
5 CR cos cf>0 -f L co sin d>0) H ---- - - *-r- R
2^
(cl>0-a)). (13)
Remembering that 0 is small, and that L is generally small
compared with Gg> we find as a sufficiently approximate value of R,
766.] The resistance is thus determined in electromagnetic mea
sure in terms of the velocity co and the deviation </>. It is not
necessary to determine H, the horizontal terrestrial magnetic force,
provided it remains constant during the experiment.
M
To determine — we must make use of the suspended magnet to
deflect the magnet of the magnetometer, as described in Art. 454.
In this experiment M should be small, so that this correction be
comes of secondary importance.
For the other corrections required in this experiment see the
Report of tli e British Association for 1863, p. 168.
767.] JOULE'S METHOD. 367
Joule's Calorimetric Method.
767.] The heat generated by a current y in passing through a
conductor whose resistance is R is, by Joule's law, Art. 242.
(1)
where / is the equivalent in dynamical measure of the unit of heat
employed.
Hence, if R is constant during the experiment, its value is
(2)
This method of determining R involves the determination of ^,
the heat generated by the current in a given time, and of y2, the
square of the strength of the current.
In Joule's experiments *, h was determined by the rise of tem
perature of the water in a vessel in which the conducting wire was
immersed. It was corrected for the effects of radiation, &c. by
alternate experiments in which no current was passed through the
wire.
The strength of the current was measured by means of a tangent
galvanometer. This method involves the determination of the
intensity of terrestrial magnetism, which was done by the method
described in* Art. 457. These measurements were also tested by the
current weigher, described in Art. 726, which measures y2 directly.
The most direct method of measuring / y2 dty however, is to pass
the current through a self-acting electrodynamometer (Art. 725)
with a scale which gives readings proportional to y2, and to make
the observations at equal intervals of time, which may be done
approximately by taking the reading at the extremities of every
vibration of the instrument during the whole course of the experi
ment.
* Report of the British Association for 1867.
CHAPTER XIX.
COMPARISON OF THE ELECTROSTATIC WITH THE ELECTRO
MAGNETIC UNITS.
Determination of the Number of Electrostatic Units of Electricity
in one Electromagnetic Unit.
768.] THE absolute magnitudes of the electrical units in both
systems depend on the units of length, time, and mass which we
adopt, and the mode in which they depend on these units is
different in the two systems, so that the ratio of the electrical units
will be expressed by a different number, according to the different
units of length and time.
It appears from the table of dimensions, Art. 628, that the
number of electrostatic units of electricity in one electromagnetic
unit varies inversely as the magnitude of the unit of length, and
directly as the magnitude of the unit of time which we adopt.
If, therefore, we determine a velocity which is represented nu
merically by this number, then, even if we adopt new units of
length and of time, the number representing this velocity will still
be the number of electrostatic units of electricity in one electro
magnetic unit, according to the new system of measurement.
This velocity, therefore, which indicates the relation between
electrostatic and electromagnetic phenomena, is a natural quantity
of definite magnitude, and the measurement of this quantity is one
of the most important researches in electricity.
To shew that the quantity we are in search of is really a velocity,
we may observe that in the case of two parallel currents the attrac
tion experienced by a length a of one of them is, by Art. 686,
F=
o
where (7, C' are the numerical values of the currents in electromag-
769.] 11ATIO EXPRESSED BY A VELOCITY. 369
netic measure, and I the distance between them. If we make
b = 2 a, then p _ CC\
Now the quantity of electricity transmitted by the current C in
the time t is Ct in electromagnetic measure, or nCt in electrostatic
measure, if n is the number of electrostatic units in one electro
magnetic unit.
Let two small conductors be charged with the quantities of
electricity transmitted by the two currents in the time t, and
placed at a distance r from each other. The repulsion between
them will be CC'n2t2
F'= —72-
Let the distance r be so chosen that this repulsion is equal to the
attraction of the currents, then
Hence r = nt-,
or the distance r must increase with the time t at the rate n.
Hence n is a velocity, the absolute magnitude of which is the
same, whatever units we assume.
769.] To obtain a physical conception of this velocity, let us ima
gine a plane surface charged with electricity to the electrostatic sur
face-density <r, and moving in its own plane with a velocity v. This
moving electrified surface will be equivalent to an electric current-
sheet, the strength of the current flowing through unit of breadth
of the surface being- av in electrostatic measure, or - av in elec-
n
tromagnetic measure, if n is the number of electrostatic units in
one electromagnetic unit. If another plane surface, parallel to the
first, is electrified to the surface-density o-', and moves in the same
direction with the velocity v', it will be equivalent to a second
current-sheet.
The electrostatic repulsion between the two electrified surfaces is,
by Art. 124, 2 ir<r<r' for every unit of area of the opposed surfaces.
The electromagnetic attraction between the two current-sheets
is, by Art. 653, 2 ituu' for every unit of area, u and u' being the
surface-densities of the currents in electromagnetic measure.
But u = - (TV. and u' = - </v', so that the attraction is
n n
,vv'
27TO-0- — jr.
n2
VOL. II. B b
370 COMPARISON OF UNITS. [770.
The ratio of the attraction to the repulsion is equal to that of
vvf to n2. Hence, since the attraction and the repulsion are quan
tities of the same kind, n must be a quantity of the same kind as v,
that is, a velocity. If we now suppose the velocity of each of the
moving planes to be equal to %, the attraction will be equal to the
repulsion, and there will be no mechanical action between them.
Hence we may define the ratio of the electric units to be a velocity,
such that two electrified surfaces, moving in the same direction
with this velocity, have no mutual action. Since this velocity is
about 288000 kilometres per second, it is impossible to make the
experiment above described.
770.] If the electric surface-density and the velocity can be made
so great that the magnetic force is a measurable quantity, we may
at least verify our supposition that a moving electrified body is
equivalent to an electric current.
It appears from Art. 57 that an electrified surface in air would
begin to discharge itself by sparks when the electric force 2 TTO-
reaches the value 130. The magnetic force due to the current-sheet
v
is 2 TTCT - • The horizontal magnetic force in Britain is about 0.175.
n
Hence a surface electrified to the highest degree, and moving with
a velocity of 100 metres per second, would act on a magnet with a
force equal to about one-four-thousandth part of the earth's hori
zontal force, a quantity which can be measured. The electrified
surface may be that of a non-conducting disk revolving in the plane
of the magnetic meridian, and the magnet may be placed close to
the ascending or descending portion of the disk, and protected from
its electrostatic action by a screen of metal. I am not aware that
this experiment has been hitherto attempted.
I. Comparison of Units of Electricity.
771.] Since the ratio of the electromagnetic to the electrostatic
unit of electricity is represented by a velocity, we shall in future
denote it by the symbol v. The first numerical determination of
this velocity was made by Weber and Kohlrausch *.
Their method was founded on the measurement of the same
quantity of electricity, first in electrostatic and then in electro
magnetic measure.
The quantity of electricity measured was the charge of a Leyden
jar. It was measured in electrostatic measure as the product of the
* Elektrodynamische Maasbestimmungen ; and Pogg. Ann. xcix, (Aug. 10, 1856.)
77I-] METHOD OF WEBER AND KOHLRAUSCH. 371
capacity of the jar into the difference of potential of its coatings.
The capacity of the jar was determined by comparison with that of
a sphere suspended in an open space at a distance from other
bodies. The capacity of such a sphere is expressed in electrostatic
measure by its radius. Thus the capacity of the jar may be found
and expressed as a certain length. See Art. 227.
The difference of the potentials of the coatings of the jar was mea
sured by connecting the coatings with the electrodes of an electro
meter, the constants of which were carefully determined, so that the
difference of the potentials, U, became known in electrostatic measure.
By multiplying this by c, the capacity of the jar, the charge of
the jar was expressed in electrostatic measure.
To determine the value of the charge in electromagnetic measure,
the jar was discharged through the coil of a galvanometer. The
effect of the transient current on the magnet of the galvanometer
communicated to the magnet a certain angular velocity. The
magnet then swung round to a certain deviation, at which its
velocity was entirely destroyed by the opposing action of the
earth's magnetism.
By observing the extreme deviation of the magnet the quantity
of electricity in the current may be determined in electromagnetic
measure, as in Art. 748, by the formula
// T
Q = -^ - 2 sin i<9,
where Q is the quantity of electricity in electromagnetic measure.
We have therefore to determine the following quantities : —
U, the intensity of the horizontal component of terrestrial mag
netism ; see Art. 456.
G, the principal constant of the galvanometer; see Art. 700.
T, the time of a single vibration of the magnet ; and
6, the deviation due to the transient current.
The value of v obtained by MM. Weber and Kohlrausch was
v — 310740000 metres per second.
The property of solid dielectrics, to which the name of Electric
Absorption has been given, renders it difficult to estimate correctly
the capacity of a Ley den jar. The apparent capacity varies ac
cording to the time which elapses between the charging or dis
charging of the jar and the measurement of the potential, and the
longer the time the greater is the value obtained for the capacity of
the jar.
B b 2
372 COMPARISON OF UNITS. [772.
Hence, since the time occupied in obtaining1 a reading of the
electrometer is large in comparison with the time during which the
discharge through the galvanometer takes place, it is probable that
the estimate of the discharge in electrostatic measure is too high,
and the value of v, derived from it, is probably also too high.
II. v expressed as a Resistance,
772. J Two other methods for the determination of v lead to an
expression of its value in terms of the resistance of a given con
ductor, which, in the electromagnetic system, is also expressed as a
velocity.
In Sir William Thomson's form of the experiment, a constant
current is made to flow through a wire of great resistance. The
electromotive force which urges the current through the wire is mea
sured electrostatically by connecting the extremities of the wire with
the electrodes of an absolute electrometer, Arts. 217, 218. The
strength of the current in the wire is measured in electromagnetic
measure by the deflexion of the suspended coil of an electrodyna-
mometer through which it passes, Art. 725. The resistance of the
circuit is known in electromagnetic measure by comparison with a
standard coil or Ohm. By multiplying the strength of the current
by this resistance we obtain the electromotive force in electro
magnetic measure, and from a comparison of this with the electro
static measure the value of v is obtained.
This method requires the simultaneous determination of two
forces, by means of the electrometer and electrodynamometer re
spectively, and it is only the ratio of these forces which appears in
the result.
773.] Another method, in which these forces, instead of being
separately measured, are directly opposed to each other, was em
ployed by the present writer. The ends of the great resistance coil
are connected with two parallel disks, one of which is moveable.
The same difference of potentials which sends the current through
the great resistance, also causes an attraction between these disks.
At the same time, an electric current which, in the actual experi
ment, was distinct from the primary current, is sent through two
coils, fastened, one to the back of the fixed disk, and the other to
the back of the moveable disk. The current flows in opposite
directions through these coils, so that they repel one another. By
adjusting the distance of the two disks the attraction is exactly
balanced by the repulsion, while at the same time another observer,
774-] METHODS OF THOMSON AND MAXWELL. 373
by means of a differential galvanometer with shunts, determines
the ratio of the primary to the secondary current.
In this experiment the only measurement which must he referred
to a material standard is that of the great resistance, which must
be determined in absolute measure by comparison with the Ohm.
The other measurements are required only for the determination of
ratios, and may therefore be determined in terms of any arbitrary
unit.
Thus the ratio of the two forces is a ratio of equality.
The ratio of the two currents is found by a comparison of resist
ances when there is no deflexion of the differential galvanometer.
The attractive force depends on the square of the ratio of the
diameter of the disks to their distance.
The repulsive force depends on the ratio of the diameter of the
coils to their distance.
The value of v is therefore expressed directly in terms of the
resistance of the great coil, which is itself compared with the Ohm.
The value oft?, as found by Thomson's method, was 28.2 Ohms* ;
by Maxwell's, 28.8 Ohmsf.
III. Electrostatic Capacity in Electromagnetic Measure.
774.] The capacity of a condenser may be ascertained in electro
magnetic measure by a comparison of the electromotive force which
produces the charge, and the quantity of electricity in the current
of discharge. By means of a voltaic battery a current is maintained
through a circuit containing a coil of great resistance. The con
denser is charged by putting its electrodes in contact with those of
che resistance coil. The current through the coil is measured by
the deflexion which it produces in a galvanometer. Let $ be this
deflexion, then the current is, by Art. 742,
H
TT = — tan <f>,
where H is the horizontal component of terrestrial magnetism, and
G is the principal constant of the galvanometer.
If R is the resistance of the coil through which this current is
made to flow, the difference of the potentials at the ends of the
coil is E= R-y,
* Report of British Association, 1869, p. 434.
t Phil. Trans., 1868, p. 643; and Report of British Association, 1869, p. 436.
374 COMPAKISON OF UNITS. [775.
and the charge of electricity produced in the condenser, whose
capacity in electromagnetic measure is C, will he
Now let the electrodes of the condenser, and then those of the
galvanometer, be disconnected from the circuit,, and let the magnet
of the galvanometer be brought to rest at its position of equili
brium. Then let the electrodes of the condenser be connected with
those of the galvanometer. A transient current will flow through
the galvanometer, and will cause the magnet to swing to an ex
treme deflexion 0. Then, by Art. 748, if the discharge is equal to
the charge, jj f
Q= 2sini0.
(JT 7T
We thus obtain as the value of the capacity of the condenser in
electromagnetic measure
C——— 2sin^
TT It tan <p
The capacity of the condenser is thus determined in terms of the
following quantities : —
Tt the time of vibration of the magnet of the galvanometer from
rest to rest.
R, the resistance of the coil.
0, the extreme limit of the swing produced by the discharge.
<£, the constant deflexion due to the current through the coil ~R.
This method was employed by Professor Fleeming Jenkin in deter
mining the capacity of condensers in electromagnetic measure *.
If c be the capacity of the same condenser in electrostatic mea
sure, as determined by comparison with a condenser whose capacity
can be calculated from its geometrical data,
c = v*C.
tan$
Hence v2
T 2 sm
The quantity v may therefore be found in this way. It depends
on the determination of R in electromagnetic measure, but as it
involves only the square root of JR, an error in this determination
will not affect the value of v so much as in the method of Arts.
772, 773.
Intermittent Current.
775.] If the wire of a battery-circuit be broken at any point, and
* Report of British Association, 1867.
776.] WIPPE. 375
the broken ends connected with the electrodes of a condenser, the
current will flow into the condenser with a strength which dimin
ishes as the difference of the potentials of the condenser increases,
so that when the condenser has received the full charge corre
sponding to the electromotive force acting on the wire the current
ceases entirely.
If the electrodes of the condenser are now disconnected from the
ends of the wire, and then again connected with them in the
reverse order, the condenser will discharge itself through the wire,
and will then become recharged in the opposite way, so that a
transient current will flow through the wire, the total quantity of
which is equal to two charges of the condenser.
By means of a piece of mechanism (commonly called a Commu
tator, or wippe] the operation of reversing the connexions of the
condenser can be repeated at regular intervals of time, each interval
being equal to T. If this interval is sufficiently long to allow of
the complete discharge of the condenser, the quantity of electricity
transmitted by the wire in each interval will be 2 EC, where E is
the electromotive force, and C is the capacity of the condenser.
If the magnet of a galvanometer included in the circuit is loaded,
so as to swing so slowly that a great many discharges of the con
denser occur in the time of one free vibration of the magnet, the
succession of discharges will act on the magnet like a steady current
whose strength is 2 EC
~~T~
If the condenser is now removed, and a resistance coil substituted
for it, and adjusted till the steady current through the galvano
meter produces the same deflexion as the succession of discharges,
and if E is the resistance of the whole circuit when this is the case,
E _2EC.
~R- ~T~
R = TC- (2)
We may thus compare the condenser with its commutator in
motion to a wire of a certain electrical resistance, and we may make
use of the different methods of measuring resistance described in
Arts. 345 to 357 in order to determine this resistance.
776.] For this purpose we may substitute for any one of the
wires in the method of the Differential Galvanometer, Art. 346, or
in that of Wheatstone's Bridge, Art. 347, a condenser with its com
mutator. Let us suppose that in either case a zero deflexion of the
376 COMPARISON OF UNITS. [777.
galvanometer has been obtained, first with the condenser and com
mutator, and then with a coil of resistance RL in its place, then
T
the quantity —^ will be measured by the resistance of the circuit of
2 L>
which the coil Rl forms part, and which is completed by the re
mainder of the conducting system including the battery. Hence
the resistance, R, which we have to calculate, is equal to R1, that
of the resistance coil, together with R2, the resistance of the re
mainder of the system (including the battery), the extremities of
the resistance coil being taken as the electrodes of the system.
In the cases of the differential galvanometer and Wheatstone's
Bridge it is not necessary to make a second experiment by substi
tuting a resistance coil for the condenser. The value of the resist
ance required for this purpose may be found by calculation from
the other known resistances in the system.
Using the notation of Art. 347, and supposing the condenser
and commutator substituted for the conductor AC in Wheatstone's
Bridge, and the galvanometer inserted in OA, and that the deflexion
of the galvanometer is zero, then we know that the resistance of a
coil, which placed in AC would give a zero deflexion, is
* = J = *!• (3)
The other part of the resistance, R2, is that of the system of con
ductors AO, OC, AB} BC and OB, the points A and C being con
sidered as the electrodes. Hence
R - ^(g
In this expression a denotes the internal resistance of the battery
and its connexions, the value of which cannot be determined with
certainty ; but by making it small compared with the other resist
ances, this uncertainty will only slightly affect the value of R2 .
The value of the capacity of the condenser in electromagnetic
measure is ^
=
777.] If the condenser has a large capacity, and the commutator
is very rapid in its action, the condenser may not be fully discharged
at each reversal. The equation of the electric current during the
discharge is
+SC = 0, (6)
where Q is the charge, C the capacity of the condenser, R2 the
778.] CONDENSER COMPARED WITH COIL. 377
resistance of the rest of the system between the electrodes of the
condenser, and E the electromotive force due to the connexions
with the battery.
Hence Q = (QQ + EC)e~W-EC, (7)
where Q0 is the initial value of Q.
If T is the time during which contact is maintained during each
discharge, the quantity in each discharge is
\+e
By making c and y in equation (4) large compared with ft, a, or
a, the time represented by R2C may be made so small compared
with r, that in calculating the value of the exponential expression
we may use the value of C in equation (5). We thus find
- Ol (9)
RJG" ~^T~ T9
where R± is the resistance which must be substituted for the con
denser to produce an equivalent effect. R2 is the resistance of the
rest of the system, T is the interval between the beginning of a
discharge and the beginning of the next discharge, and r is the
duration of contact for each discharge. We thus obtain for the
corrected value of C in electromagnetic measure
l+e *2 T
~
- 71 rri
\—e Rz T
IV. Comparison of the Electrostatic Capacity of a Condenser with
the Electromagnetic Capacity of Self-induction of a Coil.
778.] If two points of a conducting
circuit, between which the resistance is
R, are connected with the electrodes of
a condenser whose capacity is (7, then,
when an electromotive force acts on the
circuit, part of the current, instead of
passing through the resistance R, will
be employed in charging the condenser.
The current through R will therefore
rise to its final value from zero in a
gradual manner. It appears from the
mathematical theory that the manner in which the current through
378 COMPARISON OF UNITS. [77^.
R rises from zero to its final value is expressed by a formula of
exactly the same kind as that which expresses the value of a cur
rent urged by a constant electromotive force through the coil of an
electromagnet. Hence we may place a condenser and an electro
magnet on two opposite members of Wheatstone's Bridge in such
a way that the current through the galvanometer is always zero,
even at the instant of making or breaking the battery circuit.
In the figure, let P, Q, R, S be the resistances of the four mem
bers of Wheatstone's Bridge respectively. Let a coil, whose coeffi
cient of self-induction is It, be made part of the member AH, whose
resistance is Q, and let the electrodes of a condenser, whose capacity
is C, be connected by pieces of small resistance with the points F
and Z. For the sake of simplicity, we shall assume that there is no
current in the galvanometer G, the electrodes of which are con
nected to F and //. We have therefore to determine the condition
that the potential at F may be equal to that at H. It is only when
we wish to estimate the degree of accuracy of the method that we
require to calculate the current through the galvanometer when
this condition is not fulfilled.
Let x be the total quantity of electricity which has passed
through the member AF, and z that which has passed through FZ
at the time t, then x — z will be the charge of the condenser. The
electromotive force acting between the electrodes of the condenser
is, by Ohm's law, R — , so that if the capacity of the condenser
. (i)
Let y be the total quantity of electricity which has passed through
the member AH, the electromotive force from A to H must be equal
to that from A to F, or
Since there is no current through the galvanometer, the quantity
which has passed through HZ must be also y, and we find
8% = X* (3)
dt dt
Substituting in (2) the value of x, derived from (1), and com
paring with (3), we find as the condition of no current through the
galvanometer
779-] CONDENSER COMBINED WITH COIL. 379
The condition of no final current is, as in the ordinary form of
Wheatstone's Bridge, Qff _ $p (5)
The condition of no current at making and breaking the battery
connexion is r
± = RC. (6)
Here -~ and RC are the time-constants of the members Q and R
respectively, and if, by varying Q or R, we can adjust the members
of Wheatstone's Bridge till the galvanometer indicates no current,
either at making and breaking the circuit, or when the current is
steady, then we know that the time-constant of the coil is equal to
that of the condenser.
The coefficient of self-induction, L> can be determined in electro
magnetic measure from a comparison with the coefficient of mutual
induction of two circuits, whose geometrical data are known
(Art. 756). It is a quantity of the dimensions of a line.
The capacity of the condenser can be determined in electrostatic
measure by comparison with a condenser whose geometrical data
are known (Art. 229). This quantity is also a length, c. The elec
tromagnetic measure of the capacity is
Substituting this value in equation (8), we obtain for the value
of v2
v* = j QR, (8)
where c is the capacity of the condenser in electrostatic measure,
L the coefficient of self-induction of the coil in electromagnetic
measure, and Q and R the resistances in electromagnetic measure.
The value of v, as determined by this method, depends on the
determination of the unit of resistance, as in the second method,
Arts. 772, 773.
V. Combination of the Electrostatic Capacity of a Condenser with
the Electromagnetic Capacity of Self-induction of a Coil.
779.] Let C be the capacity of the condenser, the surfaces of
which are connected by a wire of resistance R. In this wire let the
coils L and L' be inserted, and let L denote the sum of their ca
pacities of self-induction. The coil L' is hung by a bifilar suspen
sion, and consists of two coils in vertical planes, between which
380
COMPARISON OF UNITS.
[779-
passes a vertical axis which carries the magnet M, the axis of which
revolves in a horizontal plane between the coils L' L. The coil L
has a large coefficient of self-induction, and is fixed. The sus
pended coil IS is protected from the
currents of air caused by the rota
tion of the magnet by enclosing the
rotating parts in a hollow case.
The motion of the magnet causes
currents of induction in the coil, and
these are acted on by the magnet,
so that the plane of the suspended
coil is deflected in the direction of
the rotation of the magnet. Let
us determine the strength of the
induced currents, and the magnitude
of the deflexion of the suspended
coil.
Let x be the charge of electricity
on the upper surface of the condenser C, then, if E is the electro
motive force which produces this charge, we have, by the theory of
the condenser, x — CE. (1)
We have also, by the theory of electric currents,
d
= 0,
(2)
where M is the electromagnetic momentum of the circuit L', when
the axis of the magnet is normal to the plane of the coil,, and 6 is
the angle between the axis of the magnet and this normal.
The equation to determine x is therefore
-n
+CR-- +>==
at
-
at
(3)
If the coil is in a position of equilibrium, and if the rotation of
the magnet is uniform, the angular velocity being »,
6 = wt. (4)
The expression for the current consists of two parts, one of which
is independent of the term on the right-hand of the equation,
and diminishes according to an exponential function of the time.
The other, which may be called the forced current, depends entirely
on the term in 0, and may be written
x = A sin 0 + £ cos 0. (5)
779-] CONDENSER COMBINED WITH COIL. 381
Finding the values of A and B by substitution, in the equation (3),
we obtain RCn cos6-(l-CLn2)sm9
The moment of the force with which the magnet acts on the coil
L' ', in which the current x is flowing, is
0 = x~(Mcos0) = Jfsin*—- (7)
dQ clt
Integrating this expression with respect to t> and dividing by t,
we find, for the mean value of 0,
- 1
~ * R*
If the coil has a considerable moment of inertia, its forced vibra
tions will be very small, and its mean deflexion will be proportional
to 0.
Let D19 DD D3 be the observed deflexions corresponding to an
gular velocities nlt n2, n3 of the magnet, then in general
, (9)
D \>n
where P is a constant.
Eliminating P and R from three equations of this form, we find
/IJ
If n2 is such that CLn^ = 1, the value of -=- will be a minimum
for this value of n. The other values of n should be taken, one
greater, and the other less, than n2.
The value of CL, determined from this equation, is of the dimen
sions of the square of a time. Let us call it r2.
If C9 be the electrostatic measure of the capacity of the con
denser, and Lm the electromagnetic measure of the self-induction of
the coil, both C9 and Lm are lines, and the product
C8Lm = v*CsL8 = v*CmLm = vV ; (11)
and f!-*^, (12)
where r2 is the value of C2Z2, determined by this experiment. The
experiment here suggested as a method of determining v is of the
same nature as one described by Sir W. R. Grove, PhU. Mag.,
382 COMPARISON OF UNITS. [780.
March 1868, p. 184. See also remarks on that experiment, by the
present writer, in the number for May 1868.
VI. Electrostatic Measurement of Resistance. (See Art. 355.)
780.] Let a condenser of capacity C be discharged through a
conductor of resistance R, then, if x is the charge at any instant,
_
Hence x = xQe R°. (2)
If, by any method, we can make contact for a short time, which
is accurately known, so as to allow the current to flow through the
conductor for the time t, then, if EQ and £J1 are the readings of an
electrometer put in connexion with the condenser before and after
the operation, RC(loge E0-log, E^ = t. (3)
If C is known in electrostatic measure as a linear quantity, R
may be found from this equation in electrostatic measure as the
reciprocal of a velocity.
If Rs is the numerical value of the resistance as thus determined,
and Rm the numerical value of the resistance in electromagnetic
measure, r>
"2 =Sr (4)
Since it is necessary for this experiment that R should be very
great, and since R must be small in the electromagnetic experi
ments of Arts. 763, &c., the experiments must be made on separate
conductors, and the resistance of these conductors compared by the
ordinary methods.
CHAPTER XX.
ELECTROMAGNETIC THEORY OF LIGHT.
781.] IN several parts of this treatise an attempt has been made
to explain electromagnetic phenomena by means of mechanical
action transmitted from one body to another by means of a medium
occupying the space between them. The undulatory theory of light
also assumes the existence of a medium. We have now to shew
that the properties of the electromagnetic medium are identical with
those of the luminiferous medium.
To fill all space with a new medium whenever any new phe
nomenon is to be explained is by no means philosophical, but if
the study of two different branches of science has independently
suggested the idea of a medium, and if the properties which must
be attributed to the medium in order to account for electro
magnetic phenomena are of the same kind as those which we
attribute to the luminiferous medium in order to account for the
phenomena of light, the evidence for the physical existence of the
medium will be considerably strengthened.
But the properties of bodies are capable of quantitative measure
ment. We therefore obtain the numerical value of some property of
the medium, such as the velocity with which a disturbance is pro
pagated through it, which can be calculated from electromagnetic
experiments, and also observed directly in the case of light. If it
should be found that the velocity of propagation of electromagnetic
disturbances is the same as the velocity of light, and this not only
in air, but in other transparent media, we shall have strong reasons
for believing that light is an electromagnetic phenomenon, and the
combination of the optical with the electrical evidence will produce
a conviction of the reality of the medium similar to that which we
obtain, in the case of other kinds of matter, from the combined
evidence of the senses.
384 ELECTROMAGNETIC THEORY OF LIGHT.
782.] When light is emitted, a certain amount of energy is
expended by the luminous body, and if the light is absorbed by
another body, this body becomes heated, shewing that it has re
ceived energy from without. During the interval of time after the
light left the first body and before it reached the second, it must
have existed as energy in the intervening space.
According to the theory of emission, the transmission of energy
is effected by the actual transference of light-corpuscules from the
luminous to the illuminated body,, carrying with them their kinetic
energy, together with any other kind of energy of which they may
be the receptacles.
According to the theory of undulation, there is a material medium
which fills the space between the two bodies, and it is by the action
of contiguous parts of this medium that the energy is passed on,
from one portion to the next, till it reaches the illuminated body.
The luminiferous medium is therefore, during the passage of light
through it, a receptacle of energy. In the undulatory theory, as
developed by Huygens, Fresnel, Young, Green, &c., this energy
is supposed to be partly potential and partly kinetic. The potential
energy is supposed to be due to the distortion of the elementary
portions of the medium. We must therefore regard the medium as
elastic. The kinetic energy is supposed to be due to the vibratory
motion of the medium. We must therefore regard the medium as
having a finite density.
In the theory of electricity and magnetism adopted in this
treatise, two forms of energy are recognised, the electrostatic and
the electrokinetic (see Arts. 630 and 636), and these are supposed
to have their seat, not merely in the electrified or magnetized
bodies, but in every part of the surrounding space, where electric
or magnetic force is observed to act. Hence our theory agTees
with the undulatory theory in assuming the existence of a medium
which is capable of becoming a receptacle of two forms of energy *.
783.] Let us next determine the conditions of the propagation
of an electromagnetic disturbance through a uniform medium, which
we shall suppose to be at rest, that is, to have no motion except that
which may be involved in electromagnetic disturbances.
* ' For my own part, considering the relation of a vacuum to the magnetic force,
and the general character of magnetic phenomena external to the magnet, I am more
inclined to the notion that in the transmission of the force there is such an action,
external to the magnet, than that the effects are merely attraction and repulsion at a
distance. Such an action may be a function of the aether; for it is not at all unlikely
that, if there be an aether, it should have other uses than simply the conveyance of
radiations.' — Faraday's Experimental Researches, 3075.
783-] PROPAGATION OF ELECTROMAGNETIC DISTURBANCES. 385
Let C be the specific conductivity of the medium, K its specific
capacity for electrostatic induction, and //, its magnetic ' perme
ability.'
To obtain the general equations of electromagnetic disturbance,
we shall express the true current (£ in terms of the vector potential
$[ and the electric potential *£.
The true current (£ is made up of the conduction current $ and
the variation of the electric displacement 5), and since both of these
depend on the electromotive force (§, we find, as in Art. 611,
But since there is no motion of the medium, we may express the
electromotive force, as in Art. 599,
@ = -Sl-V*. (2)
Hence 6 =-(C+ ± K*$ (f +V*). (3)
But we may determine a relation between (£ and 51 in a different
way, as is shewn in Art. 616, the equations (4) of which may be
written 47rM(£ = V22l + V/, (4)
T dF dG dH ,M
where / = -=- + -y- -f -7- • (5)
das dy dz
Combining equations (3) and (4), we obtain
0> (6)
which we may express in the form of three equations as follows —
rf*x _„„ dJ
dy>
These are the general equations of electromagnetic disturbances.
If we differentiate these equations with respect to #, y, and z
respectively, and add, we obtain
If the medium is a non-conductor, (7=0, and V2^, which is
proportional to the volume-density of free electricity, is independent
of t. Hence / must be a linear function of ^, or a constant, or zero,
and we may therefore leave / and ^ out of account in considering
periodic disturbances.
VOL. n. re
386 ELECTROMAGNETIC THEORY OF LIGHT.
Propagation of Undulations in a Non-conducting Medium.
784.] In this case C~ 0. and the equations become
The equations in this form are similar to those of the motion of
an elastic solid, and when the initial conditions are given, the
solution can be expressed in a form given by Poisson *, and applied
by Stokes to the Theory of Diffraction f.
Let us write V = —== - (10)
If the values of F, G, H, and of -=- > -j- > — are given at every
point of space at the epoch (t — 0), then we can determine their
values at any subsequent time, t, as follows.
Let 0 be the point for which we wish to determine the value
of F at the time t. With 0 as centre, and with radius Tt, describe
a sphere. Find the initial value of J^at every point of the spherical
surface, and take the mean, F, of all these values. Find also the
•j-pi
initial values of -=- at every point of the spherical surface, and let
dF
the mean of these values be -j- •
dt
Then the value of F at the point 0, at the time t, is
Similarly G = ^(Gt)+ t-jr > \ (11)
785.] It appears, therefore, that the condition of things at the
point 0 at any instant depends on the condition of things at a
distance Vt and at an interval of time t previously, so that any
disturbance is propagated through the medium with the velocity V.
Let us suppose that when t is zero the quantities £1 and 21 are
* Mem. de I' A cad., torn, iii, p. 130.
t Cambridge Transactions, vol. ix, p. 10 (1850).
787.] VELOCITY OF LIGHT. 387
zero except within a certain space S. Then their values at 0 at
the time t will be zero, unless the spherical surface described about
0 as centre with radius Vt lies in whole or in part within the
space S. If 0 is outside the space S there will be no disturbance
at 0 until Vt becomes equal to the shortest distance from 0 to the
space S. The disturbance at 0 will then begin, and will go on till
Vt is equal to the greatest distance from 0 to any part of S. The
disturbance at 0 will then cease for ever.
786.] The quantity V, in Art. 793, which expresses the velocity
of propagation of electromagnetic disturbances in a non-conducting
medium is, by equation (9), equal to —
If the medium is air, and if we adopt the electrostatic system
of measurement, K = I and jut = - T > so that V— v, or the velocity
of propagation is numerically equal to the number of electrostatic
units of electricity in one electromagnetic unit. If we adopt the
electromagnetic system. K = — ^ and \L — 1 , so that the equation
V= v is still true.
On the theory that light is an electromagnetic disturbance, pro
pagated in the same medium through which other electromagnetic
actions are transmitted, V must be the velocity of light, a quantity
the value of which has been estimated by several methods. On the
other hand, v is the number of electrostatic units of electricity in one
electromagnetic unit, and the methods of determining this quantity
have been described in the last chapter. They are quite inde
pendent of the methods of finding the velocity of light. Hence the
agreement or disagreement of the values of Fand of v furnishes a
test of the electromagnetic theory of light.
787.] In the following table, the principal results of direct
observation of the velocity of light, either through the air or
through the planetary spaces, are compared with the principal
results of the comparison of the electric units : —
Velocity of Light (metres per second).
Fizeau 314000000
Aberration, &c., and)
Sun's Parallax ) ' '
308000000
Foucault .. .. 2983GOOOO
Ratio of Electric Units.
Weber 310740000
Maxwell ... 288000000
Thomson... 282000000.
It is manifest that the velocity of light and the ratio of the units
are quantities of the same order of magnitude. Neither of them
c c 2
388 ELECTROMAGNETIC THEORY OF LIGHT.
can be said to be determined as yet with such a degree of accuracy
as to enable us to assert that the one is greater or less than the
other. It is to be hoped that, by further experiment, the relation be
tween the magnitudes of the two quantities may be more accurately
determined.
In the meantime our theory, which asserts that these two quan
tities are equal, and assigns a physical reason for this equality, is
certainly not contradicted by the comparison of these results such
as they are.
788.] In other media than air, the velocity V is inversely pro
portional to the square root of the product of the dielectric and the
magnetic inductive capacities. According to the undulatory theory,
the velocity of light in different media is inversely proportional to
their indices of refraction.
There are no transparent media for which the magnetic capacity
differs from that of air more than by a very small fraction. Hence
the principal part of the difference between these media must depend
on their dielectric capacity. According to our theory, therefore,
the dielectric capacity of a transparent medium should be equal to
the square of its index of refraction.
But the value of the index of refraction is different for light of
different kinds, being greater for light of more rapid vibrations.
We must therefore select the index of refraction which corresponds
to waves of the longest periods, because these are the only waves
whose motion can be compared with the slow processes by which
we determine the capacity of the dielectric.
789.] The only dielectric of which the capacity has been hitherto
determined with sufficient accuracy is paraffin, for which in the solid
form M.M. Gibson and Barclay found *
K = 1.975. (12)
Dr. Gladstone has found the following values of the index of
refraction of melted paraffin, sp.g. 0.779, for the lines A, D and H : —
Temperature
54°C
A
1.4306
57°C 1.4294
D
1.4357
1.4343
H
1.4499
1.4493
from which I find that the index of refraction for waves of infinite
length would be about 1 422
The square root of K is 1.405.
The difference between these numbers is greater than can be ac-
* Phil. Trans, 1871, p. 573.
790.] PLANE WAVES. 389
counted for by errors of observation, and shews that our theories of
the structure of bodies must be much improved before we can
deduce their optical from their electrical properties. At the same
time, I think that the agreement of the numbers is such that if no
greater discrepancy were found between the numbers derived from
the optical and the electrical properties of a considerable number of
substances, we should be warranted in concluding that the square
root of 7T, though it may not be the complete expression for the
index of refraction, is at least the most important term in it.
Plane Waves.
790.] Let us now confine our attention to plane waves, the front
of which we shall suppose normal to the axis of z. All the quan
tities, the variation of which constitutes such waves, are functions
of z and t only, and are independent of x and y. Hence the equa
tions of magnetic induction, (A), Art. 591, are reduced to
dG dF
a=—-j-) b = -—> c = 0, (13)
dz dz
or the magnetic disturbance is in the plane of the wave. This
agrees with what we know of that disturbance which constitutes
light.
Putting pa, m/3 and /uty for a, b and c respectively, the equations
of electric currents, Art. 607, become
db d*F
j- = -- Y~9 '
dz dz2
da d*GL Y (14)
4 71 U U = -- j- = 9
dz dz2
4:7TfJiW = 0.
Hence the electric disturbance is also in the plane of the wave, and
if the magnetic disturbance is confined to one direction, say that of
x, the electric disturbance is confined to the perpendicular direction,
or that of y.
But we may calculate the electric disturbance in another way,
for iff, g, h are the components of electric displacement in a non
conducting medium
df dg dh
u = 7t' ' = !' " = 3r
If P, Q, R are the components of the electromotive force
-* -«• —* (16)
390
ELECTROMAGNETIC THEORY OF LIGHT.
[791.
and since there is no motion of the medium, equations (B), Art. 598,
Q = -*°, R=-d-H. (17)
become P = =- >
at
Hence u — -^-=- ,
K
K d2F
, ,
(18)
4 77 d 47T ^2
Comparing1 these values with those given in equation (14), we find
> f
(19)
J
The first and second of these equations are the equations of pro
pagation of a plane wave, and their solution is of the well-known
form F=A(z-Vt)+/2(z+n),l
o=A(*-rt)+M*+rf).\ (20)
The solution of the third equation is
KpH=A + £t, (21)
where A and B are functions of z. H is therefore either constant
or varies directly with the time. In neither case can it take part
in the propagation of waves.
791.] It appears from this that the directions, both of the mag
netic and the electric disturbances, lie in
the plane of the wave. The mathematical
form of the disturbance therefore, agrees
with that of the disturbance which consti
tutes light, in being transverse to the di
rection of propagation.
If we suppose G — 0, the disturbance
will correspond to a plane-polarized ray of
light.
The magnetic force is in this case paral-
i ill?
lei to the axis of y and equal to — ,— , and
the electromotive force is parallel to the
dF
axis of x and equal to —
dt
The mag-
Fig. 66.
netic force is therefore in a plane perpen
dicular to that which contains the electric force.
The values of the magnetic force and of the electromotive force at
a given instant at different points of the ray are represented in Fig. 66,
793-] ENERGY AND STRESS OF RADIATION. 391
for the case of a simple harmonic disturbance in one plane. This
corresponds to a ray of plane-polarized light, but whether the plane
of polarization corresponds ta the plane of the magnetic disturbance,
or to the plane of the electric disturbance, remains to be seen. See
Art. 797.
Energy and Stress of Radiation.
79.2.] The electrostatic energy per unit of volume at any point of
the wave in a non-conducting medium is
K, KdF
•i
1 /' p — _ P2 _
2/ 877 8 77 dt
(22)
The electrokinetic energy at the same point is
(23)
8 77 877/X
In virtue of equation (8) these two expressions are equal, so that at
every point of the wave the intrinsic energy of the medium is half
electrostatic and half electrokinetic.
Let j9 be the value of either of these quantities, that is, either the
electrostatic or the electrokinetic energy per unit of volume, then,
in virtue of the electrostatic state of the medium, there is a tension
whose magnitude is jo, in a direction parallel to #, combined with a
pressure, also equal to^, parallel to y and z. See Art. 107.
In virtue of the electrokinetic state of the medium there is a
tension equal to p in a direction parallel to y, combined with a
pressure equal to p in directions parallel to x and z. See Art. 643.
Hence the combined effect of the electrostatic and the electro-
kinetic stresses is a pressure equal to 2p in the direction of the
propagation of the wave. Now 2/> also expresses the whole energy
in unit of volume.
Hence in a medium in which waves are propagated there is a
pressure in the direction normal to the waves, and numerically
equal to the energy in unit of volume.
793.] Thus, if in strong sunlight the energy of the light which
falls on one square foot is 83.4 foot pounds per second, the mean
energy in one cubic foot of sunlight is about 0.0000000882 of a foot
pound, and the mean pressure on a square foot is 0.0000000882 of a
pound weight. A flat body exposed to sunlight would experience
this pressure on its illuminated side only, and would therefore be
repelled from the side on which the light falls. It is probable that
a much greater energy of radiation might be obtained by means of
392 ELECTROMAGNETIC THEORY OF LIGHT. [794.
the concentrated rays of the electric lamp. Such rays falling- on a
thin metallic disk, delicately suspended in a vacuum, might perhaps
produce an observable mechanical effect. When a disturbance of
any kind consists of terms involving sines or cosines of angles
which vary with the time, the maximum energy is double of the
mean energy. Hence, if P is the maximum electromotive force,
and /3 the maximum magnetic force which are called into play
during the propagation of light,
JET
— P2 = — /32 = mean energy in unit of volume. (24)
8 7T 8 77
With Pouillet's data for the energy of sunlight, as quoted by
Thomson, Trans. R.S.E., 1854, this gives in electromagnetic mea
sure
P = 60000000, or about 600 Darnell's cells per metre ;
/3 = 0.193, or rather more than a tenth of the horizontal mag
netic force in Britain.
Propagation of a Plane Wave in a Crystallized Medium.
794.] In calculating, from data furnished by ordinary electro
magnetic experiments, the electrical phenomena which would result
from periodic disturbances, millions of millions of which occur in a
second, we have already put our theory to a very severe test, even
when the medium is supposed to be air or vacuum. But if we
attempt to extend our theory to the case of dense media, we become
involved not only in all the ordinary difficulties of molecular theories,
but in the deeper mystery of the relation of the molecules to the
electromagnetic medium.
To evade these difficulties, we shall assume that in certain media
the specific capacity for electrostatic induction is different in dif
ferent directions, or in other words, the electric displacement, in
stead of being in the same direction as the electromotive force, and
proportional to it, is related to it by a system of linear equations
similar to those given in Art. 297. It may be shewn, as in
Art. 436, that the system of coefficients must be symmetrical, so
that, by a proper choice of axes, the equations become
f=~K,P, ff = ±X,Q, * = ±KtR, (1)
where Kl , K2 , and K3 are the principal inductive capacities of the
medium. The equations of propagation of disturbances are therefore
796.] DOUBLE REFRACTION. 393
^F__^G^ d*H (d*F d2*
~df^~dz*~ ~dx~dy dz~dx ~ 1/X \ dt2 ~ dxdt
d2F ,d2G d2*
dz2 dxz dy dz dxdy 2/^^2 dydt
d2F d2G ,d2ff d2*
dx2 dy2 dzdx dydz r\dt2 dzdt}
795.] If I, m, n are the direction-cosines of the normal to the
wave-front, and V the velocity of the wave, and if
Ix + my + nz—~Pt = w, (3)
and if we write F", G", H", V" for the second differential coeffi
cients of F, G, //, ^ respectively with respect to w, and put
1 1 1
(4)
where a, £, c are the three principal velocities of propagation, the
equations become
n*-F"-lmG"-nlH"--rV' = 0,
-ImF" +'(n2 + 1*- ~G"-mnH"- VV = 0, (5)
-nlF"- mn G" + (l2 + m2 -
796.J If we write
72
we obtain from these equations
rU(PF"-W) = 0,)
(7)
Hence, either V = 0, in which case the wave is not propagated at
all ; or, U = 0, which leads to the equation for V given by Fresnel ;
or the quantities within brackets vanish, in which case the vector
whose components are F", G" ', H" is normal to the wave-front and
proportional to the electric volume-density. Since the medium is
a non-conductor, the electric density at any given point is constant,
and therefore the disturbance indicated by these equations is not
periodic, and cannot constitute a wave. We may therefore consider
*"= 0 in the investigation of the wave.
394 ELECTROMAGNETIC THEORY OF LIGHT. [797.
797.] The velocity of the propagation of the wave is therefore
completely determined from the equation U = 0, or
I2 m2 n2 . }
7* -a2 + T*^JP + F2-c2 =
There are therefore two, and only two, values of V2 correspondiDg
to a given direction of wave-front.
If A, jot, v are the direction-cosines of the electric current whose
components are uy v, w>
A:M:,:::G":-", (9)
then l\ + mn + nv=0; (10)
or the current is in the plane of the wave-front, and its direction
in the wave-front is determined by the equation
l-(b2-c2} + ™(c*-a*)+-(a*-6*) = 0. (11)
A )U V
These equations are identical with those given by Fresnel if we
define the plane of polarization as a plane through the ray per
pendicular to the plane of the electric disturbance.
According to this electromagnetic theory of double refraction the
wave of normal disturbance, which constitutes one of the chief
difficulties of the ordinary theory, does not exist, and no new
assumption is required in order to account for the fact that a ray
polarized in a principal plane of the crystal is refracted in the
ordinary manner *.
Relation between Electric Conductivity and Opacity.
798.] If the medium, instead of being a perfect insulator, is a
conductor whose conductivity per unit of volume is C, the dis
turbance will consist not only of electric displacements but of
currents of conduction, in which electric energy is transformed into
heat, so that the undulation is absorbed by the medium.
If the disturbance is expressed by a circular function, we may
write -»t-qz), (1)
for this will satisfy the equation
,0v
'
provided q2-pz = ^Kn2, (3)
and 2p = 1-ny.Cn. (4)
* See Stokes' 'Report on Double Refraction' ; Brit. Assoc. Reports, 1862, p. 255.
8oi.] CONDUCTIVITY AND OPACITY. 395
The velocity of propagation is
r=A (5)
2
and the coefficient of absorption is
p = 27T/ACT. (6)
Let R be the resistance, in electromagnetic measure, of a plate
whose length is /, breadth #, and thickness z,
*=-se- (7)
The proportion of the incident light which will be transmitted by
this plate will be
i_v_
e-*p*=. e rMb B. (8)
799.] Most transparent solid bodies are good insulators, and all
good conductors are very opaque. There are, however, many ex
ceptions to the law that the opacity of a body is the greater, the
greater its conductivity.
Electrolytes allow an electric current to pass, and yet many of
them are transparent. We may suppose, however, that in the case
of the rapidly alternating forces which come into play during the
propagation of light, the electromotive force acts for so short a
time in one direction that it is unable to effect a complete separation
between the combined molecules. When, during the other half of
the vibration, the electromotive force acts in the opposite direction
it simply reverses what it did during the first half. There is thus
no true conduction through the electrolyte, no loss of electric
energy, and consequently no absorption of light.
800.] Gold, silver, and platinum are good conductors, and yet,
when formed into very thin plates, they allow light to pass through
them. From experiments which I have made on a piece of gold
leaf, the resistance of which was determined by Mr. Hockin, it
appears that its transparency is very much greater than is con
sistent with our theory, unless we suppose that there is less loss
of energy when the electromotive forces are reversed for every semi-
vibration of light than when they act for sensible times, as in our
ordinary experiments.
801.] Let us next ' consider the case of a medium in which the
conductivity is large in proportion to the inductive capacity.
In this case we may leave out the term involving K in the equa
tions of Art. 783, and they then become
396 ELECTROMAGNETIC THEORY OF LIGHT. [802.
(1)
Each of these equations is of the same form as the equation of the
diffusion of heat given in Fourier's Traite de Chaleur.
802.] Taking the first as an example, the component F of the
vector-potential will vary according to time and position in the same
way as the temperature of a homogeneous solid varies according
to time and position, the initial and the surface-conditions being
made to correspond in the two cases, and the quantity 47r/u,Cbeing
numerically equal to the reciprocal of the thermometric conductivity
of the substance, that is to say, the number of units of volume of
the substance which would be heated one degree by the heat which passes
through a unit cube of the substance, two opposite faces of which differ
by one degree of temperature, while the other faces are impermeable to
heat*.
The different problems in thermal conduction, of which Fourier
has given the solution, may be transformed into problems in the
diffusion of electromagnetic quantities, remembering that F, G, H
are the components of a vector, whereas the temperature, in Fourier's
problem, is a scalar quantity.
Let us take one of the cases of which Fourier has given a com
plete solution t, that of an infinite medium, the initial state t)f
which is given.
The state of any point of the medium at the time t is found
by taking the average of the state of every part of the medium,
the weight assigned to each part in taking the average being
where r is the distance of that part from the point considered. This
average, in the case of vector-quantities, is most conveniently taken
by considering each component of the vector separately.
* See Maxwell's Theory of Heat, p. 235.
t Traite de la Chalewr, Art. 384. The equation which determines the temperature,
v, at a point (x, y, z) after a time t, in terms of /(a, 0, 7), the initial temperature at
the point (0,0,7), is
r C r do. d@ dy — ( I
v=/// r=- —e * *M 'J (**&»'
/// 23\/^v3t3
«/ j j
where k is the thermometric conductivity.
804.] ESTABLISHMENT OF THE DISTRIBUTION OF FORCE. 397
803.] We have to remark in the first place, that in this problem
the thermal conductivity of Fourier's medium is to be taken in
versely proportional to the electric conductivity of our medium,
so that the time required in order to reach an assigned stage in
the process of diffusion is greater the higher the electric conduct
ivity. This statement will not appear paradoxical if we remember
the result of Art. 655, that a medium of infinite conductivity forms
a complete barrier to the process of diffusion of magnetic force.
In the next place, the time requisite for the production of an
assigned stage in the process of diffusion is proportional to the square
of the linear dimensions of the system.
There is no determinate velocity which can be defined as the
velocity of diffusion. If we attempt to measure this velocity by
ascertaining the time requisite for the production of a given amount
of disturbance at a given distance from the origin of disturbance,
we find that the smaller the selected value of the disturbance the
greater the velocity will appear to be, for however great the distance,
and however small the time, the value of the disturbance will differ
mathematically from zero.
This peculiarity of diffusion distinguishes it from wave-propaga
tion, which takes place with a definite velocity. No disturbance
takes place at a given point till the wave reaches that point, and
when the wave has passed, the disturbance ceases for ever.
804.] Let us now investigate the process which takes place when
an electric current begins and continues to flow through a linear
circuit, the medium surrounding the circuit being of finite electric
conductivity. (Compare with Art. 660).
When the current begins, its first effect is to produce a current
of induction in the parts of the medium close to the wire. The
direction of this current is opposite to that of the original current,
and in the first instant its total quantity is equal to that of the
original current, so that the electromagnetic effect on more distant
parts of the medium is initially zero, and only rises to its final
value as the induction-current dies away on account of the electric
resistance of the medium.
But as the induction-current close to the wire dies away, a new
induction -current is generated in the medium beyond, so that the
space occupied by the induction-current is continually becoming
wider, while its intensity is continually diminishing.
This diffusion and decay of the induction-current is a pheno
menon precisely analogous to the diffusion of heat from a part of
398 ELECTROMAGNETIC THEORY OF LIGHT. [805.
the medium initially hotter or colder than the rest. We must
remember, however, that since the. current is a vector quantity.,
and since in a circuit the current is in opposite directions at op
posite points of the circuit, we must, in calculating any given com
ponent of the induction-current, compare the problem with one
in which equal quantities of heat and of cold are diffused from
neighbouring places, in which case the effect on distant points will
be of a smaller order of magnitude.
805.] If the current in the linear circuit is maintained constant,
the induction currents, which depend on the initial change of state,
will gradually be diffused and die away, leaving the medium in its
permanent state, which is analogous to the permanent state of the
flow of heat. In this state we have
V2I<7 = V2£ = y2#=0 (2)
throughout the medium, except at the part occupied by the circuit,
in which V2F= 4wM ,
V2£=47r^,> (3)
V2//=477^J
These equations are sufficient to determine the values of F, G, R
throughout the medium. They indicate that there are no currents
except in the circuit, and that the magnetic forces are simply those
due to the current in the circuit according to the ordinary theory.
The rapidity with which this permanent state is established is so
great that it could not be measured by our experimental methods,
except perhaps in the case of a very large mass of a highly con
ducting medium such as copper.
NOTE. — In a paper published in PoggendorfFs Annalen, June 1867,
M. Lorenz has deduced from Kirchhoff 's equations of electric cur
rents (Pogg. Ann. cii. 1856), by the addition of certain terms which
do not affect any experimental result, a new set of equations, indi
cating that the distribution of force in the electromagnetic field
may be conceived as arising from the mutual action of contiguous
elements, and that waves, consisting of transverse electric currents,
may be propagated, with a velocity comparable to that of light, in
non-conducting media. He therefore regards the disturbance which
constitutes light as identical with these electric currents, and he
shews that conducting media must be opaque to such radiations.
These conclusions are similar to those of this chapter, though
obtained by an entirely different method. The theory given in
this chapter was first published in the PUL Trans, for 1865.
CHAPTER XXI.
MAGNETIC ACTION ON LIGHT.
806.] THE most important step in establishing a relation between
electric and magnetic phenomena and those of light must be the
discovery of some instance in which the one set of phenomena is
aifected by the other. In the search for such phenomena we must
be guided by any knowledge we may have already obtained with
respect to the mathematical or geometrical form of the quantities
which we wish to compare. Thus, if we endeavour, as Mrs. Somer-
ville did, to magnetize a needle by means of light, we must re
member that the distinction between magnetic north and south is
a mere matter of direction, and would be at once reversed if we
reverse certain conventions about the use of mathematical signs.
There is nothing in magnetism analogous to those phenomena of
electrolysis which enable us to distinguish positive from negative
electricity, by observing that oxygen appears at one pole of a cell
and hydrogen at the other.
Hence we must not expect that if we make light fall on one end
of a needle, that end will become a pole of a certain name, for the
two poles do not differ as light does from darkness.
We might expect a better result if we caused circularly polarized
light to fall on the needle, right-handed light falling on one end
and left-handed on the other, for in some respects these kinds of
light may be said to be related to each other in the same way as
the poles of a magnet. The analogy, however, is faulty even here,
for the two rays when combined do not neutralize each other, but
produce a plane polarized ray.
Faraday, who was acquainted with the method of studying the
strains produced in transparent solids by means of polarized light,
made many experiments in hopes of detecting some action on polar
ized light while passing through a medium in which electrolytic
conduction or dielectric induction exists *. He was not, however,
* Experimental Researches, 951-954 and 2216-2220.
400 MAGNETIC ACTION ON LIGHT. [807.
able to detect any action of this kind, though the experiments were
arranged in the way best adapted to discover effects of tension,
the electric force or current being at right angles to the direction
of the ray, and at an angle of forty-five degrees to the plane of
polarization. Faraday varied these experiments in many ways with
out discovering any action on light due to electrolytic currents or
to static electric induction.
He succeeded, however, in establishing a relation between light
and magnetism, and the experiments by which he did so are de
scribed in the nineteenth series of his Experimental Researches. We
shall take Faraday's discovery as our starting point for further
investigation into the nature of magnetism, and we shall therefore
describe the phenomenon which he observed.
807.] A ray of plane-polarized light is transmitted through a
transparent diamagnetic medium, and the plane of its polarization,
when it emerges from the medium, is ascertained by observing the
position of an analyser when it cuts off the ray. A magnetic force
is then made to act so that the direction of the force within the
transparent medium coincides with the direction of the ray. The
light at once reappears, but if the analyser is turned round through
a certain angle, the light is again cut off. This shews that the
effect of the magnetic force is to turn the plane of polarization,
round the direction of the ray as an axis, through a certain angle,
measured by the angle through which the analyser must be turned
in order to cut off the light.
808.] The angle through which the plane of polarization is
turned is proportional —
(1) To the distance which the ray travels within the medium.
Hence the plane of polarization changes continuously from its posi
tion at incidence to its position at emergence.
(2) To the intensity of the resolved part of the magnetic force in
the direction of the ray.
(3) The amount of the rotation depends on the nature of the
medium. No rotation has yet been observed when the medium is
air or any other gas.
These three statements are included in the more general one,
that the angular rotation is numerically equal to the amount by
which the magnetic potential increases, from the point at which
the ray enters the medium to that at which it leaves it, multiplied
by a coefficient, which, for diamagnetic media, is generally positive.
809.] In diamagnetic substances, the direction in which the plane
8io.] FARADAY'S DISCOVERY. 401
of polarization is made to rotate is the same as the direction in which
a positive current must circulate round the ray in order to produce
a magnetic force in the same direction as that which actually exists
in the medium.
Verdet, however, discovered that in certain ferromagnetic media,
as, for instance, a strong solution of perchloride of iron in wood-
spirit or ether, the rotation is in the opposite direction to the current
which would produce the magnetic force.
This shews that the difference between ferromagnetic and dia
magnetic substances does not arise merely from the ' magnetic per
meability' being in the first case greater, and in the second less,
than that of air, but that the properties of the two classes of bodies
are really opposite.
The power acquired by a substance under the action of magnetic
force of rotating the plane of polarization of light is not exactly
proportional to its diamagnetic or ferromagnetic magnetizability.
Indeed there are exceptions to the rule that the rotation is positive for
diamagnetic and negative for ferromagnetic substances, for neutral
chromate of potash is diamagnetic, but produces a negative rotation.
810.] There are other substances, which, independently of the
application of magnetic force, cause the plane of polarization to
turn to the right or to the left, as the ray travels through the sub
stance. In some of these the property is related to an axis, as in
the case of quartz. In others, the property is independent of the
direction of the ray within the medium, as in turpentine, solution
of sugar, &c. In all these substances, however, if the plane of
polarization of any ray is twisted within the medium like a right-
handed screw, it will still be twisted like a right-handed screw if
the ray is transmitted through the medium in the opposite direction.
The direction in which the observer has to turn his analyser in order
to extinguish the ray after introducing the medium into its path,
is the same with reference to the observer whether the ray comes
to him from the north or from the south. The direction of the
rotation in space is of course reversed when the direction of the ray is
reversed. But when the rotation is produced by magnetic action, its
direction in space is the same whether the ray be travelling north
or south. The rotation is always in the same direction as that of
the electric current which produces, or would produce, the actual
magnetic state of the field, if the medium belongs to the positive
class, or in the opposite direction if the medium belongs to the
negative class.
VOL. IT. D d
402 MAGNETIC ACTION ON LIGHT. [8 1 I.
It follows from this, that if the ray of light, after passing through
the medium from north to south, is reflected by a mirror, so as to
return through the medium from south to north,, the rotation will
be doubled when it results from magnetic action. When the rota
tion depends on the nature of the medium alone, as in turpentine, &c.,
the ray, when reflected back through the medium, emerges in the
same plane as it entered, the rotation during the first passage
through the medium having been exactly reversed during the
second.
811.] The physical explanation of the phenomenon presents con
siderable difficulties, which can hardly be said to have been hitherto
overcome, either for the magnetic rotation, or for that which
certain media exhibit of themselves. We may, however, prepare
the way for such an explanation by an analysis of the observed
facts.
It is a well-known theorem in kinematics that two uniform cir
cular vibrations, of the same amplitude, having the same periodic
time, and in the same plane, but revolving in opposite directions,
are equivalent, when compounded together, to a rectilinear vibra
tion. The periodic time of this vibration is equal to that of the
circular vibrations, its amplitude is double, and its direction is in
the line joining the points at which two particles, describing' the
circular vibrations in opposite directions round the same circle,
would meet. Hence if one of the circular vibrations has its phase
accelerated, the direction of the rectilinear vibration will be turned,
in the same direction as that of the circular vibration, through an
angle equal to half the acceleration of phase.
It can also be proved by direct optical experiment that two rays
of light, circularly-polarized in opposite directions, and of the same
intensity, become, when united, a plane-polarized ray, and that if
by any means the phase of one of the circularly-polarized rays is
accelerated, the plane of polarization of the resultant ray is turned
round half the angle of acceleration of the phase.
812.] We may therefore express the phenomenon of the rotation
of the plane of polarization in the following manner : — A plane-
polarized ray falls on the medium. This is equivalent to two cir
cularly-polarized rays, one right-handed, the other left-handed (as
regards the observer) . After passing through the medium the ray
is still plane-polarized, but the plane of polarization is turned, say,
to the right (as regards the observer) . Hence, of the two circularly-
polarized rays, that which is right-handed must have had its phase
8 14-]
STATEMENT OF THE FACTS.
403
accelerated with respect to the other during its passage through the
medium.
In other words, the right-handed ray has performed a greater
number of vibrations, and therefore has a smaller wave-length,
within the medium, than the left-handed ray which has the same
periodic time.
This mode of stating what takes place is quite independent of
any theory of light, for though we use such terms as wave-length,
circular-polarization, &c., which may be associated in our minds
with a particular form of the undulatory theory, the reasoning is
independent of this association, and depends only on facts proved
by experiment.
813.] Let us next consider the configuration of one of these rays
at a given instant. Any undulation, the motion of which at each
point is circular, may be represented by a helix or screw. If the
screw is made to revolve about its axis without any longitudinal
motion, each particle will describe a circle, and at the same time the
propagation of the undulation will be represented by the apparent
longitudinal motion of the similarly situated parts of the thread of
the screw. It is easy to see that if the screw is right-handed, and
the observer is placed at that end towards which the undulation
travels, the motion of the screw will appear to him left-handed,
that is to say, in the opposite di
rection to that of the hands of a
watch. Hence such a ray has
been called, originally by French
writers, but now by the whole
scientific world, a left-handed cir
cularly-polarized ray.
A right-handed circularly-polar
ized ray is represented in like
manner by a left-handed helix.
In Fig. 67 the right-handed helix
A, on the right-hand of the figure,
represents a left-handed ray, and
the left-handed helix B, on the left-
hand, represents a right-handed
ray.
814.] Let us now consider two
such rays which have the same
wave-length within the medium.
67<
They are geometrically alike in
B d i
404 MAGNETIC ACTION OX LIGHT. [815.
all respects, except that one is the perversion of the other, like its
image in a looking-glass. One of them, however, say A, has a
shorter period of rotation than the other. If the motion is entirely
due to the forces called into play by the displacement, this shews
that greater forces are called into play by the same displacement
when the configuration is like A than when it is like B. Hence in
this case the left-handed ray will be accelerated with respect to the
right-handed ray, and this will be the case whether the rays are
travelling from N to S or from S to N.
This therefore is the explanation of the phenomenon as it is pro
duced by turpentine, &c. In these media the displacement caused
by a circularly-polarized ray calls into play greater forces of resti
tution when the configuration is like A than when it is like B.
The forces thus depend on the configuration alone, not on the direc
tion of the motion.
But in a diamagnetic medium acted on by magnetism in the
direction SN9 of the two screws A and B, that one always rotates
with the greatest velocity whose motion, as seen by an eye looking
from S to N, appears like that of a watch. Hence for rays from S
to N the right-handed ray B will travel quickest, but for rays
from N to 8 the left-handed ray A will travel quickest.
815.] Confining our attention to one ray only, the helix B has
exactly the same configuration, whether it represents a ray from S
to N or one from N to S. But in the first instance the ray travels
faster, and therefore the helix rotates more rapidly. Hence greater
forces are called into play when the helix is going round one way
than when it is going round the other way. The forces, therefore,
do not depend solely on the configuration of the ray, but also on
the direction of the motion of its individual parts.
816.] The disturbance which constitutes light, whatever its
physical nature may be, is of the nature of a vector, perpendicular
to the direction of the ray. This is proved from the fact of the
interference of two rays of light, which under certain conditions
produces darkness, combined with the fact of the non-interference
of two rays polarized in planes perpendicular to each other. For
since the interference depends on the angular position of the planes
of polarization, the disturbance must be a directed quantity or
vector, and since the interference ceases when the planes of polar
ization are at right angles, the vector representing the disturbance
must be perpendicular to the line of intersection of these planes,
that is, to the direction of the ray.
817.] C1KCULARLY-POLAKIZED LIGHT. 405
817.] The disturbance, being a vector, can be resolved into com
ponents parallel to x and y, the axis of z being4 parallel to the
direction of the ray. Let f and 77 be these components, then, in the
case of a ray of homogeneous circularly-polarized light,
f = rcosO, rj = rsmO, (1)
where 0 = nt — qz + a. (2)
In these expressions, r denotes the magnitude of the vector, and
0 the angle which it makes with the direction of the axis of x.
The periodic time, r, of the disturbance is such that
UT — 27T. (3)
The wave-length, A, of the disturbance is such that
q\ = 27T. (4)
The velocity of propagation is - •
The phase of the disturbance when t and z are both zero is a.
The circularly-polarized light is right-handed or left-handed
according as q is negative or positive.
Its vibrations are in the positive or the negative direction of
rotation in the plane of (no, y}^ according as n is positive or negative.
The light is propagated in the positive or the negative direction
of the axis of z, according as n and q are of the same or of opposite
signs.
In all media n varies when q varies, and -=- is always of the same
sign with - •
Hence, if for a given numerical value of n the value of - is
greater when n is positive than when n is negative, it follows that
for a value of q, given both in magnitude and sign, the positive
value of n will be greater than the negative value.
Now this is what is observed in a diamagnetic medium, acted on
by a magnetic force, y, in the direction of z. Of the two circularly-
polarized rays of a given period, that is accelerated of which the
direction of rotation in the plane of (#, y) is positive. Hence, of
two circularly-polarized rays, both left-handed, whose wave-length
within the medium is the same, that has the shortest period whose
direction of rotation in the plane of xy is positive, that is, the ray
which is propagated in the positive direction of z from south to
north. We have therefore to account for the fact, that when in the
equations of the system q and r are given, two values of n will
406 MAGNETIC ACTION ON LIGHT. [8 1 8.
satisfy the equations, one positive and the other negative, the
positive value being numerically greater than the negative.
818.] We may obtain the equations of motion from a considera
tion of the potential and kinetic energies of the medium. The
potential energy, F, of the system depends on its configuration,
that is, on the relative position of its parts. In so far as it depends
on the disturbance due to circularly-polarized light, it must be a
function of r, the amplitude, and q, the coefficient of torsion, only.
It may be different for positive and negative values of q of equal
numerical value, and it probably is so in the case of media which
of themselves rotate the plane of polarization.
The kinetic energy, T, of the system is a homogeneous function
of the second degree of the velocities of the system, the coefficients
of the different terms being functions of the coordinates.
819.] Let us consider the dynamical condition that the ray may
be of constant intensity, that is, that r may be constant.
Lagrange's equation for the force in r becomes
d dT dT
Since r is constant, the first term vanishes. We have therefore the
equation dT dV . .
' Tr + ~dr = ' ( '
in which q is supposed to be given, and we are to determine the
value of the angular velocity 0, which we may denote by its actual
value, n.
The kinetic energy, T, contains one term involving n2 ; other
terms may contain products of n with other velocities, and the
rest of the terms are independent of n. The potential energy, T7, is
entirely independent of n. The equation is therefore of the form
An* + Bn+C = 0. (7)
This being a quadratic equation, gives two values of n. It appears
from experiment that both values are real, that one is positive and
the other negative, and that the positive value is numerically the
greater. Hence, if A is positive, both B and C are negative, for,
if % and n2 are the roots of the equation,
^(% + O + -#=0. (8)
The coefficient, _Z?, therefore, is not zero, at least when magnetic
force acts on the medium. We have therefore to consider the ex
pression Bn, which is the part of the kinetic energy involving the
first power of n, the angular velocity of the disturbance.
821.] MAGNETISM IMPLIES AN ANGULAR TELOCITY. 407
820.] Every term of T is of two dimensions as regards velocity.
Hence the terms involving- n must involve some other velocity.
This velocity cannot be r or q, because, in the case we consider,
r and q are constant. Hence it is a velocity which exists in the
medium independently of that motion which constitutes light. It
must also be a velocity related to n in such a way that when it is
multiplied by n the result is a scalar quantity, for only scalar quan
tities can occur as terms in the value of T, which is itself scalar.
Hence this velocity must be in the same direction as n, or in the
opposite direction, that is, it must be an angular velocity about the
axis of z.
Again, this velocity cannot be independent of the magnetic force,
for if it were related to a direction fixed in the medium, the phe
nomenon would be different if we turned the medium end for end,
which is not the case.
We are therefore led to the conclusion that this velocity is an
invariable accompaniment of the magnetic force in those media
which exhibit the magnetic rotation of the plane of polarization.
8.21.] We have been hitherto obliged to use language which is
perhaps too suggestive of the ordinary hypothesis of motion in the
undulatory theory. It is easy, however, to state our result in a
form free from this hypothesis.
Whatever light is, at each point of space there is something
going on, whether displacement, or rotation, or something not yet
imagined, but which is certainly of the nature of a vector or di
rected quantity, the direction of which is normal to the direction
of the ray. This is completely proved by the phenomena of inter
ference.
In the case of circularly-polarized light, the magnitude of this
vector remains always the same, but its direction rotates round the
direction of the ray so as to complete a revolution in the periodic
time of the wave. The uncertainty which exists as to whether this
vector is in the plane of polarization or perpendicular to it, does not
extend to our knowledge of the direction in which it rotates in right-
handed and in left-handed circularly-polarized light respectively.
The direction and the angular velocity of this vector are perfectly
known, though the physical nature of the vector and its absolute
direction at a given instant are uncertain.
When a ray of circularly-polarized light falls on a medium under
the action of magnetic force, its propagation within the medium
is affected by the relation of the direction of rotation of the light to
408 MAGNETIC ACTION ON LIGHT. [822.
the direction of the magnetic force. From this we conclude, by the
reasoning of Art. 821, that in the medium, when under the action
of magnetic force, some rotatory motion is going on, the axis of ro
tation being in the direction of the magnetic forces ; and that the
rate of propagation of circularly-polarized light, when the direction
of its vibratory rotation and the direction of the magnetic rotation
of the medium are the same, is different from the rate of propaga
tion when these directions are opposite.
The only resemblance which we can trace between a medium
through which circularly-polarized light is propagated, and a me
dium through which lines of magnetic force pass, is that in both
there is a motion of rotation about an axis. But here the resem
blance stops, for the rotation in the optical phenomenon is that of
the vector which represents^ the disturbance. This vector is always
perpendicular to the direction of the ray, and rotates about it a
known number of times in a second. In the magnetic phenomenon,
that which rotates has no properties by which its sides can be dis
tinguished, so that we cannot determine how many times it rotates
in a second.
There is nothing, therefore, in the magnetic phenomenon which
corresponds to the wave-length and the wave-propagation in the op
tical phenomenon. A medium in which a constant magnetic force
is acting is not, in consequence of that force, filled with waves
travelling in one direction, as when light is propagated through it.
The only resemblance between the optical and the magnetic pheno
menon is, that at each point of the medium something exists of
the nature of an angular velocity about an axis in the direction of
the magnetic force.
On the Hypothesis of Molecular Vortices.
822.] The consideration of the action of magnetism on polarized
light leads, as we have seen, to the conclusion that in a medium
under the action of magnetic force something belonging to the
same mathematical class as an angular velocity, whose axis is in the
direction of the magnetic force, forms a part of the phenomenon.
This angular velocity cannot be that of any portion of the me
dium of sensible dimensions rotating as a whole. We must there
fore conceive the rotation to be that of very small portions of the
medium, each rotating on its own axis. This is the hypothesis of
molecular vortices.
The motion of these vortices, though, as we have shewn (Art. 575),
824.] MOLECULAR VOHTICES. 409
it does not sensibly affect the visible motions of large bodies, may
be such as to affect that vibratory motion on which the propagation
of light, according to the undulatory theory, depends. The dis
placements of the medium, during the propagation of light, will
produce a disturbance of the vortices, and the vortices when so dis
turbed may react on the medium so as to affect the mode of propa
gation of the ray.
823.] It is impossible, in our present state of ignorance as to the
nature of the vortices, to assign the form of the law which connects
the displacement of the medium with the variation of the vortices.
We shall therefore assume that the variation of the vortices caused
by the displacement of the medium is subject to the same conditions
which Helmholtz, in his great memoir on Vortex-motion *, has
shewn to regulate the variation of the vortices of a perfect liquid.
Helmholtz's law may be stated as follows : — Let P and Q be two
neighbouring particles in the axis of a vortex, then, if in conse
quence of the motion of the fluid these particles arrive at the
points P'Q', the line P'Q' will represent the new direction of the
axis of the vortex, and its strength will be altered in the ratio of
P'Q' to PQ.
Hence if a, /3, y denote the components of the strength of a vor
tex, and if f, 17, f denote the displacements of the medium, the value
of a will become
/ d£ 0 d^ d£ ^
a = a + a -= — f-p -= — |-y -y- >
ax ay dz
We now assume that the same condition is satisfied during the
small displacements of a medium in which a, (3, y represent, not
the components of the strength of an ordinary vortex, but the
components of magnetic force.
824.] The components of the angular velocity of an element of
the medium are Wl = \ — (*£ - ^?) , ]
dt V dy dz '
(2)
* Crelle's Journal, vol. Iv. (1858). Translated by Tait, Phil. Mag., July, 1867.
410 MAGNETIC ACTION ON LIGHT. [825-
The next step in our hypothesis is the assumption that the
kinetic energy of the medium contains a term of the form
2<?(ao>1 + /3a>2 + y6>3). (3)
This is equivalent to supposing that the angular velocity acquired
by the element of the medium during the propagation of light is a
quantity which may enter into combination with that motion by
which magnetic phenomena are explained.
In order to form the equations of motion of the medium, we must
express its kinetic energy in terms of the velocity of its parts,
the components of which are f, 77, f We therefore integrate by
parts, and find
2 C 1 1 1 (acoj + /3a>2 -f ya>3) dx dy dz
+ cff(aC- yfl dz dx + OJJ(ft- arj) dx dy
The double integrals refer to the bounding surface, which may be
supposed at an infinite distance. We may, therefore, while in
vestigating what takes place in the interior of the medium, confine
our attention to the triple integral.
825.] The part of the kinetic energy in unit of volume, expressed
by this triple integral, may be written
**C(t»+iiv + tw), (5)
where u, v, w are the components of the electric current as given in
equations (E), Art. 607.
It appears from this that our hypothesis is equivalent to the
assumption that the velocity of a particle of the medium whose
components are f, r/, £ is a quantity which may enter into com
bination with the electric current whose components are u, v, w.
826.] Returning to the expression under the sign of triple inte
gration in (4), substituting for the values of a, ft, y, those of
a', /3', /, as given by equations (1), and writing
d d d d
the expression under the sign of integration becomes
dr d ,d d d sdr
dk zdh Tz "" r/ dk dx dy'
In the case of waves in planes normal to the axis of z the displace-
828.] DYNAMICAL THEOEY. 411
ments are functions of z and t only, so that -77 = y -j- > and this
dfi dz
expression is reduced to
^
The kinetic energy per unit of volume, so far as it depends on
the velocities of displacement, may now be written
where p is the density of the medium.
827.] The components, X and Y9 of the impressed force, referred
to unit of volume, may be deduced from this by Lagrange's equa
tions, Art. 564.
' (10)
<»>
These forces arise from the action of the remainder of the medium
on the element under consideration, and must in the case of an
isotropic medium be of the form indicated by Cauchy,
828.] If we now take the case of a circularly-polarized ray for
which f = rcos(nt—qz), r] = r sin (nt - qz\ (14)
we find for the kinetic energy in unit of volume
T — \pr*n2 — Cyr2q*n', (15)
and for the potential energy in unit of volume
= r*Q, (16)
where Q is a function of q2.
The condition of free propagation of the ray given in Art. 820,
equation (6), is dT _dV
dr dr
which gives Pn2-2Cyq2n = Q, (18)
whence the value of n may be found in terms of q.
But in the case of a ray of given wave-period, acted on by
412 MAGNETIC ACTION ON LIGHT. [829.
magnetic force, what we want to determine is the value of —-, when n
is constant, in terms of ~ , when y is constant. Differentiating (1 8)
(2pn — 2Cyf)dn— {-j^ + lCygnjdti—'ZCifndy = 0. (19)
We thus find -f = - —^ ~f • (2°)
ay pn—Cyq2 an
829.] If A is the wave-length in air, and i the corresponding
index of refraction in the medium,
q\ = 2ni, n\ = 2irv. (21)
The change in the value of q, due to magnetic action, is in every
case an exceedingly small fraction of its own value, so that we may
^ %, (22)
where qt0 is the value of q when the magnetic force is zero. The
angle, 0, through which the plane of polarization is turned in
passing through a thickness c of the medium, is half the sum of
the positive and negative values of qc, the sign of the result being
changed, because the sign of q is negative in equations (14). We
thus obtain
0=-cy^ (23)
4 TT C i2 . di x 1
The second term of the denominator of this fraction is approx
imately equal to the angle of rotation of the plane of polarization
during its passage through a thickness of the medium equal to half
a wave-length. It is therefore in all actual cases a quantity which
we may neglect in comparison with unity.
Writing ~ = m, (25)
vp
we may call m the coefficient of magnetic rotation for the medium,
a quantity whose value must be determined by observation. It is
found to be positive for most diamagnetic, and negative for some
paramagnetic media. We have therefore as the final result of our
theory *2 j;
'-x, (26)
where 6 is the angular rotation of the plane of polarization, m a
830.] FORMULA FOR THE ROTATION. 413
constant determined by observation of the medium, y the intensity
of the magnetic force resolved in the direction of the ray, c the
length of the ray within the medium, X the wave-length of the
light in air, and i its index of refraction in the medium.
830.] The only test to which this theory has hitherto been sub
jected, is that of comparing the values of 0 for different kinds of
light passing through the same medium and acted on by the same
magnetic force.
This has been done for a considerable number of media by M.
Verdet "*, who has arrived at the following results : —
(1) The magnetic rotations of the planes of polarization of the
rays of different colours follow approximately the law of the inverse
square of the wave-length.
(2) The exact law of the phenomena is always such that the pro
duct of the rotation by the square of the wave-length increases from
the least refrangible to the most refrangible end of the spectrum.
(3) The substances for which this increase is most sensible are
also those which have the greatest dispersive power.
He also found that in the solution of tartaric acid, which of itself
produces a rotation of the plane of polarization, the magnetic rotation
is by no means proportional to the natural rotation.
In an addition to the same memoir f Verdet has given the results
of very careful experiments on bisulphide of carbon and on creosote,
two substances in which the departure from the law of the inverse
square of the wave-length was very apparent. He has also com
pared these results with the numbers given by three different for
mulae, f'2 Jj .
(i) 0-.
(II) e -.
(ill) e-.
w/v
The first of these formulae, (I), is that which we have already ob
tained in Art. 829, equation (26). The second, (II), is that which
results from substituting in the equations of motion, Art. 826, equa-
70 70 >> -70
cL t\ cL * f] YI
tions (10), (11), terms of the form -~ and —-j^, instead of -=-5-3-
cl/t dt dz dt
* Recherches sur leg proprie'te's optiques de'veloppe'es dans les corps transparents
par Faction du magn^tisme, 4me partie. Comptes JfawfttS, t. Ivi. p. 630 (6 April, 1863).
t Comptes Rendw, Ivii. p. 670 (19 Oct., 1863).
414 MAGNETIC ACTION ON LIGHT. [830.
and -- j-A • I am no^ aware that this form of the equations has
dz^dt
been suggested by any physical theory. The third formula, (III),
results from the physical theory of M. C. Neumann *, in which the
equations of motion contain terms of the form ~ and — -- t.
dt dt
It is evident that the values of 6 given by the formula (III) are
not even approximately proportional to the inverse square of the
wave-length. Those given by the formulae (I) and (II) satisfy this
condition, and give values of 6 which agree tolerably well with the
observed values for media of moderate dispersive power. For bisul
phide of carbon and creosote, however, the values given by (II) differ
very much from those observed. Those given by (I) agree better
with observation, but, though the agreement is somewhat close for
bisulphide of carbon, the numbers for creosote still differ by quan
tities much greater than can be accounted for by any errors of
observation.
Magnetic Rotation of the Plane of Polarization (from Verdef).
Bisulphide of Carbon at 24°. 9 C.
Lines of the spectrum C D E F G
Observed rotation 592 768 1000 1234 1704
Calculated by I. 589 760 1000 1234 1713
II. 606 772 1000 1216 1640
III. 943 967 1000 1034 1091
Rotation of the ray E = 25°. 28'.
Creosote at 24°. 3 C.
Lines of the spectrum C D E F 0
Observed rotation 573 758 1000 1241 1723
Calculated by I. 617 780 1000 1210 1603
II. 623 789 1000 1200 1565
III. 976 993 1000 1017 1041
Rotation of the ray E = 21°. 58'.
We are so little acquainted with the details of the molecular
* ' Explicare tentatur quomodo fiat ut lucis planum polarizationis per vires elec-
tricas vel magneticas declinetur.' Halis Saxonum, 1858.
•f* These three forms of the equations of motion were first suggested by Sir G. B.
Airy (Phil. Mag., June 1846) as a means of analysing the phenomenon then recently
discovered by Faraday. Mac Cullagh had previously suggested equations containing
terms of the form — in order to represent mathematically the phenomena of quartz.
These equations were offered by Mac Cullagh and Airy, 'not as giving a mechanical
explanation of the phenomena, but as shewing that the phenomena may be explained
by equations, which equations appear to be such as might possibly be deduced from
some plausible mechanical assumption, although no such assumption lias yet been
made.'
831.] ARGUMENT OF THOMSON. 415
constitution of bodies, that it is not probable that any satisfactory
theory can be formed relating to a particular phenomenon, such as
that of the magnetic action on light, until, by an induction founded
on a number of different cases in which visible phenomena are found
to depend upon actions in which the molecules are concerned, we
learn something more definite about the properties which must be
attributed to a molecule in order to satisfy the conditions of ob
served facts.
The theory proposed in the preceding pages is evidently of a
provisional kind, resting as it does on unproved hypotheses relating
to the nature of molecular vortices, and the mode in which they are
affected by the displacement of the medium. We must therefore
regard any coincidence with observed facts as of much less scientific
value in the theory of the magnetic rotation of the plane of polari
zation than in the electromagnetic theory of light, which, though it
involves hypotheses about the electric properties of media, does not
speculate as to the constitution of their molecules.
831.] NOTE. — The whole of this chapter may be regarded as an
expansion of the exceedingly important remark of Sir William
Thomson in the Proceedings of the Royal Society, June 1856 : — ' The
magnetic influence on light discovered by Faraday depends on the
direction of motion of moving particles. For instance, in a medium
possessing it, particles in a straight line parallel to the lines of
magnetic force, displaced to a helix round this line as axis, and then
projected tangentially with such velocities as to describe circles,
will have different velocities according as their motions are round
in one direction (the same as the nominal direction of the galvanic
current in the magnetizing coil), or in the contrary direction. But
the elastic reaction of the medium must be the same for the same
displacements, whatever be the velocities and directions of the par
ticles ; that is to say, the forces which are balanced by centrifugal
force of the circular motions are equal, while the luminiferous
motions are unequal. The absolute circular motions being there
fore either equal or such as to transmit equal centrifugal forces to
the particles initially considered, it follows that the luminiferous
motions are only components of the whole motion ; and that a less
luminiferous component in one direction, compounded with a mo
tion existing in the medium when transmitting no light, skives an
equal resultant to that of a greater luminiferous motion in the con
trary direction compounded with the same non -luminous motion.
I think it is not only impossible to conceive any other than this
410 MAGNETIC ACTION ON LIGHT.
dynamical explanation of the fact that circularly-polarized light
transmitted through magnetized glass parallel to the lines of mag
netizing force, with the same quality, right-handed always, or left-
handed always, is propagated at different rates according as its
course is in the direction or is contrary to the direction in which a
north magnetic pole is drawn ; but I believe it can be demonstrated
that no other explanation of that fact is possible. Hence it appears
that Faraday's optical discovery affords a demonstration of the re
ality of Ampere's explanation of the ultimate nature of magnetism ;
and gives a definition of magnetization in the dynamical theory of
heat. The introduction of the principle of moments of momenta
(" the conservation of areas ") into the mechanical treatment of
Mr. Rankine's hypothesis of " molecular vortices," appears to indi
cate a line perpendicular to the plane of resultant rotatory mo
mentum ("the invariable plane") of the thermal motions as the
magnetic axis of a magnetized body, and suggests the resultant
moment of momenta of these motions as the definite measure of
the " magnetic moment." The explanation of all phenomena of
electromagnetic attraction or repulsion, and of electromagnetic in
duction, is to be looked for simply in the inertia and pressure of
the matter of which the motions constitute heat. Whether this
matter is or is not electricity, whether it is a continuous fluid inter-
permeating the spaces between molecular nuclei, or is itself mole-
cularly grouped ; or whether all matter is continuous, and molecular
heterogeneousness consists in finite vortical or other relative mo
tions of contiguous parts of a body ; it is impossible to decide, and
perhaps in vain to speculate, in the present state of science.'
A theory of molecular vortices, which I worked out at consider
able length, was published in the Phil. Mag. for March, April, and
May, 1861, Jan. and Feb. 1862.
I think we have good evidence for the opinion that some pheno
menon of rotation is going on in the magnetic field, that this rota
tion is performed by a great number of very small portions of
matter, each rotating on its own axis, this axis being parallel to the
direction of the magnetic force, and that the rotations of these dif
ferent vortices are made to depend on one another by means of some
kind of mechanism connecting them.
The attempt which I then made to imagine a working model of
this mechanism must be taken for no more than it really is, a de
monstration that mechanism may be imagined capable of producing
a connexion mechanically equivalent to the actual connexion of the
831.] THEOBY OP MOLECULAK VORTICES. 417
parts of the electromagnetic field. The problem of determining the
mechanism required to establish a given species of connexion be
tween the motions of the parts of a system always admits of an
infinite number of solutions. Of these, some may be more clumsy
or more complex than others, but all must satisfy the conditions of
mechanism in general.
The following results of the theory, however, are of higher
value : —
(1) Magnetic force is the effect of the centrifugal force of the
vortices.
(2) Electromagnetic induction of currents is the effect of the
forces called into play when the velocity of the vortices is changing.
(3) Electromotive force arises from the stress on the connecting
mechanism.
(4) Electric displacement arises from the elastic yielding of the
connecting mechanism.
VOL. II.
CHAPTER XXII
FEBROMAQNETISM AND DIAMAGNETISM EXPLAINED BY
MOLECULAR CURRENTS.
On Electromagnetic Theories of Magnetism.
832.] WE have seen (Art. 380) that the action of magnets on
one another can be accurately represented by the attractions and
repulsions of an imaginary substance called * magnetic matter.'
We have shewn the reasons why we must not suppose this magnetic
matter to move from one part of a magnet to another through a
sensible distance, as at first sight it appears to do when we
magnetize a bar, and we were led to Poisson's hypothesis that the
magnetic matter is strictly confined to single molecules oi" the mag
netic substance, so that a magnetized molecule is one in which the
opposite kinds of magnetic matter are more or less separated to
wards opposite poles of the molecule, but so that no part of either
can ever be actually separated from the molecule (Art. 430).
These arguments completely establish the fact, that magnetiza
tion is a phenomenon, not of large masses of iron, but of molecules,
that is to say, of portions of the substance so small that we cannot
by any mechanical method cut one of them in two, so as to obtain a
north pole separate from a south pole. But the nature of a mag
netic molecule is by no means determined without further investi
gation. We have seen (Art. 442) that there are strong reasons for
believing that the act of magnetizing iron or steel does not consist
in imparting magnetization to the molecules of which it is com
posed, but that these molecules are already magnetic, even in un-
magnetized iron, but with their axes placed indifferently in all
directions, and that the act of magnetization consists in turning
the molecules so that their axes are either rendered all parallel to
one direction, or at least. are deflected towards that direction.
834-] AMPERE'S THEORY. 419
833.] Still, however, we have arrived at no explanation of the
nature of a magnetic molecule, that is, we have not recognized its
likeness to any other thing of which we know more. We have
therefore to consider the hypothesis of Ampere, that the magnetism
of the molecule is due to an electric current constantly circulating
in some closed path within it.
It is possible to produce an exact imitation of the action of any
magnet on points external to it, by means of a sheet of electric
currents properly distributed on its outer surface. But the action
of the magnet on points in the interior is quite different from the
action of the electric currents on corresponding points. Hence Am
pere concluded that if magnetism is to be explained by means of
electric currents, these currents must circulate within the molecules
of the magnet, and must not flow from one molecule to another.
As we cannot experimentally measure the magnetic action at a
point in the interior of a molecule, this hypothesis cannot be dis
proved in the same way that we can disprove the hypothesis of
currents of sensible extent within the magnet.
Besides this, we know that an electric current, in passing from
one part of a conductor to another, meets with resistance and gene
rates heat ; so that if there were currents of the ordinary kind round
portions of the magnet of sensible size, there would be a constant
expenditure of energy required to maintain them, and a magnet
would be a perpetual source of heat. By confining the circuits to
the molecules, within which nothing is known about resistance, we
may assert, without fear of contradiction, that the current, in cir
culating within the molecule, meets with no resistance.
According to Ampere's theory, therefore, all the phenomena of
magnetism are due to electric currents, and if we could make ob
servations of the magnetic force in the interior of a magnetic mole
cule, we should find that it obeyed exactly the same laws as the
force in a region surrounded by any other electric circuit.
834.] In treating of the force in the interior of magnets, we have
supposed the measurements to be made in a small crevasse hollowed
out of the substance of the magnet, Art. 395. We were thus led
to consider two different quantities, the magnetic force and the
magnetic induction, both of which are supposed to be observed in
a space from which the magnetic matter is removed. We were
not supposed to be able to penetrate into the interior of a mag
netic molecule and to observe the force within it.
If we adopt Ampere's theory, we consider a magnet, not as a
E e 2
420 ELECTE1C THEORY OF MAGNETISM. [835.
continuous substance, the magnetization of which varies from point
to point according to some easily conceived law, but as a multitude
of molecules, within each of which circulates a system of electric
currents, giving rise to a distribution of magnetic force of extreme
complexity, the direction of the force in the interior of a molecule
being generally the reverse of that of the average force in its neigh
bourhood, and the magnetic potential, where it exists at all, being
a function of as many degrees of multiplicity as there are molecules
in the magnet.
835.] But we shall find, that, in spite of this apparent complexity,
which, however, arises merely from the coexistence of a multitude
of simpler parts, the mathematical theory of magnetism is greatly
simplified by the adoption of Ampere's theory, and by extending
our mathematical vision into the interior of the molecules.
In the first place, the two definitions of magnetic force are re
duced to one, both becoming the same as that for the space outside
the magnet. In the next place, the components of the magnetic
force everywhere satisfy the condition to which those of induction
are subject, namely, da dp, dy _
dx dy dz ~
In other words, the distribution of magnetic force is of the
same nature as that of the velocity of an incompressible fluid,
or, as we have expressed it in Art. 25, the magnetic force has no
convergence.
Finally, the three vector functions — the electromagnetic momen
tum, the magnetic force, and the electric current — become more
simply related to each other. They are all vector functions of no
convergence, and they are derived one from the other in order, by
the same process of taking the space-variation, which is denoted
by Hamilton by the symbol V.
836.] But we are now considering magnetism from a physical
point of view, and we must enquire into the physical properties of
the molecular currents. We assume that a current is circulating
in a molecule, and that it meets with no resistance. If L is the
coefficient of self-induction of the molecular circuit, and M the co
efficient of mutual induction between this circuit and some other
circuit, then if y is the current in the molecule, and y that in the
other circuit, the equation of the current y is
=-Sr, (2)
838.] CIRCUITS OF NO RESISTANCE. 421
and since by the hypothesis there is no resistance, R = 0, and we
get by integration
Ly + My = constant, = Lyot say. (3)
Let us suppose that the area of the projection of the molecular
circuit on a plane perpendicular to the axis of the molecule is A,
this axis being defined as the normal to the plane on which the
projection is greatest. If the action of other currents produces a
magnetic force, X, in a direction whose inclination to the axis of
the molecule is 0, the quantity My becomes XA cos0, and we have
as the equation of the current
Ly + XAco$e — Ly0, (4)
where y0 is the value of y when X = 0.
It appears, therefore, that the strength of the molecular current
depends entirely on its primitive value y0, and on the intensity of
the magnetic force due to other currents.
837.] If we suppose that there is no primitive current, but that
the current is entirely due to induction, then
* XA
y = j— cos 0. (o)
Jj
The negative sign shews that the direction of the induced cur
rent is opposite to that of the inducing current, and its magnetic
action is such that in the interior of the circuit it acts in the op
posite direction to the magnetic force. In other words, the mole
cular current acts like a small magnet whose poles are turned
towards the poles of the same name of the inducing magnet.
Now this is an action the reverse of that of the molecules of iron
under magnetic action. The molecular currents in iron, therefore,
are not excited by induction. But in diamagnetic substances an
action of this kind is observed, and in fact this is the explanation of
diamagnetic polarity which was first given by Weber.
Weber's Theory of Diamagnetism.
838.] According to Weber's theory, there exist in the molecules
of diamagnetic substances certain channels round which an electric
current can circulate without resistance. It is manifest that if we
suppose these channels to traverse the molecule in every direction,
this amounts to making the molecule a perfect conductor.
Beginning with the assumption of a linear circuit within the mo
lecule, we have the strength of the current given by equation (5).
422 ELECTRIC THEORY OF MAGNETISM. [8 39.
The magnetic moment of the current is the product of its strength
by the area of the circuit, or yA, and the resolved part of this in the
direction of the magnetizing force is yAcosO, or, by (5),
Y //2
-^-cos20. (6)
If there are n such molecules in unit of volume, and if their axes are
distributed indifferently in all directions, then the average value of
cos20 will be J, and the intensity of magnetization of the substance
will be ^nXA* ,?.
L
Neumann's coefficient of magnetization is therefore
_
The magnetization of the substance is therefore in the opposite
direction to the magnetizing force, or, in other words, the substance
is diamagnetic. It is also exactly proportional to the magnetizing
force, and does not tend to a finite limit, as in the case of ordinary
magnetic induction. See Arts. 442, &c.
839.] If the directions of the axes of the molecular channels are
arranged, not indifferently in all directions, but with a preponder
ating number in certain directions, then the sum
Ju
extended to all the molecules will have different values according
to the direction of the line from which 6 is measured, and the dis
tribution of these values in different directions will be similar to the
distribution of the values of moments of inertia about axes in dif
ferent directions through the same point.
Such a distribution will explain the magnetic phenomena related
to axes in the body, described by Pliicker, which Faraday has called
Magne-crystallic phenomena. See Art. 435.
840.] Let us now consider what would be the effect, if, instead
of the electric current being confined to a certain channel within
the molecule, the whole molecule were supposed a perfect conductor.
Let us begin with the case of a body the form of which is acyclic,
that is to say, which is not in the form of a ring or perforated
body, and let us suppose that this body is everywhere surrounded
by a thin shell of perfectly conducting matter.
We have proved in Art. 654, that a closed sheet of perfectly
conducting matter of any form, originally free from currents, be-
842.] PERFECTLY CONDUCTING MOLECULES. 423
comes, when exposed to external magnetic force, a current-sheet, the
action of which on every point of the interior is such as to make
the magnetic force zero.
It may assist us in understanding this case if we observe that
the distribution of magnetic force in the neighbourhood of such a
body is similar to the distribution of velocity in an incompressible
fluid in the neighbourhood of an impervious body of the same form.
It is obvious that if other conducting shells are placed within
the first, since they are not exposed to magnetic force, no currents
will be excited in them. Hence, in a solid of perfectly conducting
material, the effect of magnetic force is to generate a system of
currents which are entirely confined to the surface of the body.
841.] If the conducting body is in the form of a sphere of radius
r, its magnetic moment is
and if a number of such spheres are distributed in a medium, so
that in unit of volume the volume of the conducting matter is Xf,
then, by putting ^=1, and /x2 = 0 in equation (17), Art. 314, we find
the coefficient of magnetic permeability,
f\ n If
(9)
whence we obtain for Poisson's magnetic coefficient
t=-\tf, (10)
and for Neumann's coefficient of magnetization by induction
Since the mathematical conception of perfectly conducting bodies
leads to results exceedingly different from any phenomena which
we can observe in ordinary conductors, let us pursue the subject
somewhat further.
842.] Returning to the case of the conducting channel in the
form of a closed curve of area A, as in Art. 836, we have, for the
moment of the electromagnetic force tending to increase the angle 0,
n0 m (12)
= — ^-sin0cos0. (13)
This force is positive or negative according as 0 is less or greater
than a right angle. Hence the effect of magnetic force on a per
fectly conducting channel tends to turn it with its axis at right
424 ELECTRIC THEORY OF MAGNETISM. [843.
angles to the line of magnetic force, that is, so that the plane of the
channel becomes parallel to the lines of force.
An effect of a similar kind may be observed by placing a penny
or a copper ring between the poles of an electromagnet. At the
instant that the magnet is excited the ring turns its plane towards
the axial direction, but this force vanishes as soon as the currents
are deadened by the resistance of the copper *.
843.] We have hitherto considered only the case in which the
molecular currents are entirely excited by the external magnetic
force. Let us next examine the bearing of Weber's theory of the
magneto-electric induction of molecular currents on Ampere's theory
of ordinary magnetism. According to Ampere and Weber, the
molecular currents in magnetic substances are not excited by the
external magnetic force, but are already there, and the molecule
itself is acted on and deflected by the electromagnetic action of the
magnetic force on the conducting circuit in which the current flows.
When Ampere devised this hypothesis, the induction of electric cur
rents was not known, and he made no hypothesis to account for the
existence, or to determine the strength, of the molecular currents.
We are now, however, bound to apply to these currents the same
laws that Weber applied to his currents in diamagnetic molecules.
We have only to suppose that the primitive value of the current y,
when no magnetic force acts, is not zero but y0. The strength of
the current when a magnetic force, X, acts on a molecular current
of area A, whose axis is inclined 6 to the line of magnetic force, is
and the moment of the couple tending to turn the molecule so as
to increase 0 is X2A2
— y0XAsm0 + sin 26. (15)
Hence, putting A
AyQ = m, /- = *, (16)
^7o
in the investigation in Art. 443, the equation of equilibrium becomes
Xsin0 — 3X2sin0cos0 = Dsin(a-0). (17)
The resolved part of the magnetic moment of the current in the
direction of X is
XA2
y A cosO = y0Acos0 -- ^— cos2 (9, (18)
L
= mcosO(l-3XcoaO). (19)
* See Faraday, Exp. Res., 2310, &c.
845-] MODIFIED THEORY OF INDUCED MAGNETISM. 425
844.] These conditions differ from those in Weber's theory of
magnetic induction by the terms involving the coefficient B. If
BX is small compared with unity, the results will approximate to
those of Weber's theory of magnetism. If BX is large compared
with unity, the results will approximate to those of Weber's theory
of diamagnetism.
Now the greater y0, the primitive value of the molecular current,
the smaller will B become, and if L is also large, this will also
diminish B. Now if the current flows in a ring channel, the value
T>
of L depends on log — , where R is the radius of the mean line of
the channel, and r that of its section. The smaller therefore the
section of the channel compared with its area, the greater will be L,
the coefficient of self-induction, and the more nearly will the phe
nomena agree with Weber's original theory. There will be this
difference, however, that as X, the magnetizing force, increases, the
temporary magnetic moment will not only reach a maximum, but
will afterwards diminish as X increases.
If it should ever be experimentally proved that the temporary
magnetization of any substance first increases, and then diminishes
as the magnetizing force is continually increased, the evidence of
the existence of these molecular currents would, I think, be raised
almost to the rank of a demonstration.
845.] If the molecular currents in diamagnetic substances are
confined to definite channels, and if the molecules are capable of
being deflected like those of magnetic substances, then, as the mag
netizing force increases, the diamagnetic polarity will always increase,
but, when the force is great, not quite so fast as the magnetizing
force. The small absolute value of the diamagnetic coefficient shews,
however, that the deflecting force on each molecule must be small
compared with that exerted on a magnetic molecule, so that any
result due to this deflexion is not likely to be perceptible.
If, on the other hand, the molecular currents in diamagnetic
bodies are free to flow through the whole substance of the molecules,
the diamagnetic polarity will be strictly proportional to the mag
netizing force, and its amount will lead to a determination of the
whole space occupied by the perfectly conducting masses, and, if we
know the number of the molecules, to the determination of the size
of each,
CHAPTER XXIII.
THEORIES OF ACTION AT A DISTANCE.
On the Explanation of Ampere's Formula given by Gauss and Weber.
846.] The attraction between the elements ds and da' of two
circuits, carrying electric currents of intensity i and i't is, by
Ampere's formula,
ii' ds ds' dr dr\ ft\
3--; (1)
zr_ .
r2 v ds ds ds ds '
the currents being estimated in electromagnetic units. See Art. 526.
The quantities, whose meaning as they appear in these expres
sions we have now to interpret, are
dr dr . d2r
cos e, -jr- -7-7 > and -=— T> ;
ds ds dsds
and the most obvious phenomenon in which to seek for an inter
pretation founded on a direct relation between the currents is the
relative velocity of the electricity in the two elements.
847.] Let us therefore consider the relative motion of two par
ticles, moving with constant velocities v and v' along the elements
ds and ds' respectively. The square of the relative velocity of these
particles is U2 = vz _2vv'cos e + v'2-, (3)
and if we denote by r the distance between the particles,
dr dr ,dr ...
v7 ~v 7~+v -r>> (4)
^ ds ds
.dr dr /9 /dr\2 /ev
v '' 5
848.] FECHNER'S HYPOTHESIS. 427
where the symbol <) indicates that, in the quantity differentiated,
the coordinates of the particles are to be expressed in terms of the
time.
It appears, therefore, that the terms involving the product vv' in
the equations (3), (5), and (6) contain the quantities occurring in
(1) and (2) which we have to interpret. We therefore endeavour to
~~
and — 2 • But in order to
express (1) and (2) in terms of ^2, i
do so we must get rid of the first and third terms of each of these
expressions, for they involve quantities which do not appear in the
formula of Ampere. Hence we cannot explain the electric current
as a transfer of electricity in one direction only, but we must com
bine two opposite streams in each current, so that the combined
effect of the terms involving v2 and v'2 may be zero.
848.] Let us therefore suppose that in the first element, ds, we
have one electric particle, £, moving with velocity ?;, and another, elt
moving with velocity vl , and in the same way two particles, ef and
e\, in ds't moving with velocities v' and v'L respectively.
The term involving v2 for the combined action of these particles
Similarly 2 (t/W) = (v'2e' + v\2e\) (e + ^) ; (8)
and 2(vtfeS) = (ve + v^^v'e' + vYi). (9)
In order that 2 (o2ee') may be zero, we must have either
/ + e\ = 0, or V2e + v12e1 = 0. (10)
According to Eechner's hypothesis, the electric current consists
of a current of positive electricity in the positive direction, com
bined with a current of negative electricity in the negative direc
tion, the two currents being exactly equal in numerical magnitude,
both as respects the quantity of electricity in motion and the velo
city with which it is moving. Hence both the conditions of (10)
are satisfied by Fechner's hypothesis.
But it is sufficient for our purpose to assume, either —
That the quantity of positive electricity in each element is nu
merically equal to the quantity of negative electricity ; or —
That the quantities of the two kinds of electricity are inversely
as the squares of their velocities.
Now we know that by charging the second conducting wire as a
whole, we can make e' -f e\ either positive or negative. Such a
charged wire, even without a current, according to this formula,
would act on the first wire carrying a current in which v2e -j- r12el
428 ACTION AT A DISTANCE. [849.
has a value differing from zero. Such an action has never been
observed.
Therefore, since the quantity e' + e\ may be shewn experimentally
not to be always zero, and since the quantity v2e + v21el is not
capable of being experimentally tested, it is better for these specu
lations to assume that it is the latter quantity which invariably
vanishes.
849.] Whatever hypothesis we adopt, there can be no doubt that
the total transfer of electricity, reckoned algebraically, along the
first circuit, is represented by
ve-\-v1ei = dels;
where c is the number of units of statical electricity which are
transmitted by the unit electric current in the unit of time, so that
we may write equation (9)
2 (vv'ee'} = c2 ii'ds ds'. (11)
Hence the sums of the four values of (3), (5), and (6) become
2 (ee'n2) = -2 c^ii'ds ds' cos e ; (12)
^, (13)
ds ds
and we may write the two expressions (1) and (2) for the attraction
between ds and ds'
850.] The ordinary expression, in the theory of statical electri-
PP
city, for the repulsion of two electrical particles e and e' is - , and
which gives the electrostatic repulsion between the two elements if
they are charged as wholes.
Hence, if we assume for the repulsion of the two particles either
of the modified expressions
we may deduce from them both the ordinary electrostatic forces, and
the forces acting between currents as determined by Ampere.
FORMULAE OF GAUSS AND WEBER, 429
851.] The first of these expressions, (18), was discovered by
Gauss * in July 1835, and interpreted by him as a fundamental law
of electrical action, that ' Two elements of electricity in a state of
relative motion attract or repel one another, but not in the same
way as if they are in a state of relative rest.' This discovery was
not, so far as I know, published in the lifetime of Gauss, so that the
second expression, which was discovered independently by W.Weber,
and published in the first part of his celebrated Elektrodynamische
Maasbe&timmungen^ , was the first result of the kind made known
to the scientific world.
852.] The two expressions lead to precisely the same result when
they are applied to the determination of the mechanical force be
tween two electric currents, and this result is identical with that
of Ampere. But when they are considered as expressions of the
physical law of the action between two electrical particles, we are
led to enquire whether they are consistent with other known facts
of nature.
Both of these expressions involve the relative velocity of the
particles. Now, in establishing- by mathematical reasoning the
well-known principle of the conservation of energy, it is generally
assumed that the force acting between two particles is a function of
the distance only, and it is commonly stated that if it is a function
of anything else, such as the time, or the velocity of the particles,
the proof would not hold.
Hence a law of electrical action, involving the velocity of the
particles, has sometimes been supposed to be inconsistent with the
principle of the conservation of energy.
853.] The formula of Gauss is inconsistent with this principle,
and must therefore be abandoned, as it leads to the conclusion that
energy might be indefinitely generated in a finite system by physical
means. This objection does not apply to the formula of Weber, for
he has shewn J that if we assume as the potential energy of a system
consisting of two electric particles,
the repulsion between them, which is found by differentiating this
quantity with respect to r, and changing the sign, is that given by
the formula (19).
* Werke (G-ottingen edition, 1867), \ol.v. p. 616.
t Abh. Leibnizens Qes., Leipzig (1846).
J Pogg. Ann., Ixxiii. p. 229 (1848).
430 ACTION AT A DISTANCE. [8 54.
Hence the work done on a moving particle by the repulsion of a
fixed particle is ^o~"^i' where \ITO and \//j are the values of \ff at the
beginning and at the end of its path. Now \j/ depends only on the
distance, r, and on the velocity resolved in the direction of r. If,
therefore, the particle describes any closed path, so that its position,
velocity, and direction of motion are the same at the end as at the
beginning, ^ will be equal to \^0, and no work will be done on the
whole during the cycle of operations.
Hence an indefinite amount of work cannot be generated by a
particle moving in a periodic manner under the action of the force
assumed by Weber.
854.] But Helmholtz, in his very powerful memoir on the 'Equa
tions of Motion of Electricity in Conductors at Rest '*, while he
shews that Weber's formula is not inconsistent with the principle
of the conservation of energy, as regards only the work done during
a complete cyclical operation, points out that it leads to the conclu
sion, that two electrified particles, which move according to Weber's
law, may have at first finite velocities, and yet, while still at a finite
distance from each other, they may acquire an infinite kinetic energy,
and may perform an infinite amount of work.
To this Weber f replies, that the initial relative velocity of the
particles in Helmholtz's example, though finite, is greater than the
velocity of light ; and that the distance at which the kinetic energy
becomes infinite, though finite, is smaller than any magnitude which
we can perceive, so that it may be physically impossible to bring two
molecules so near together. The example, therefore, cannot be tested
by any experimental method.
Helmholtz J has therefore stated a case in which the distances are
not too small, nor the velocities too great, for experimental verifica
tion. A fixed non-conducting spherical surface, of radius &, is uni
formly charged with electricity to the surface-density a. A particle,
of mass m and carrying a charge e of electricity, moves within the
sphere with velocity v. The electrodynamic potential calculated
from the formula (20) is
2
l-, (21)
and is independent of the position of the particle within the sphere.
Adding to this Vt the remainder of the potential energy arising
* Crelle's Journal, 72 (1870).
t Elektr. Maasl). inlmondere liber das Princip der Erhaltung der Energie.
J Ikiiin Monatslericht, April 1872; Phil May., Dec. 1872, Supp.
856.] POTENTIAL OF TWO CLOSED CURRENTS. 431
from the action of other forces, and \mv2, the kinetic energy of the
particle, we find as the equation of energy
r* const. (22)
Since the second term of the coefficient of v3 may be increased in
definitely by increasing a, the radius of the sphere, while the surface-
density a remains constant, the coefficient of v2 may be made negative.
Acceleration of the motion of the particle would then correspond to
diminution of its vis viva, and a body moving in a closed path and
acted on by a force like friction, always opposite in direction to its
motion, would continually increase in velocity, and that without
limit. This impossible result is a necessary consequence of assuming
any formula for the potential which introduces negative terms into
the coefficient of v2.
855.] But we have now to consider the application of Weber's
theory to phenomena which can be realized. We have seen how it
gives Ampere's expression for the force of attraction between two
elements of electric currents. The potential of one of these ele
ments on the other is found by taking the sum of the values of the
potential \j/ for the four combinations of the positive and negative
currents in the two elements. The result is, by equation (20), taking
the sum of the four values of ,,
di
(23)
r ds ds
and the potential of one closed current on another is
_ w /Yl d4-~ds ds' = ii' M, (24)
jj. r ds ds
i I r*o^ p
where M = 1 1 - — dsds*, as in Arts. 423, 524.
In the case of closed currents, this expression agrees with that
which we have already (Art. 524) obtained"*.
Weber s Theory of the Induction of Electric Currents.
856.] After deducing from Ampere's formula for the action
between the elements of currents, his own formula for the action
between moving electric particles, Weber proceeded to apply his
formula to the explanation of the production of electric currents by
* In the whole of this investigation Weber adopts the electrodynamic system of
units. Tn this treatise we always use the electromagnetic system. The electro-mag
netic unit of current is to the electrodynamic unit in the ratio of A/2 to 1. Art. 526.
432 ACTION AT A DISTANCE. [857.
magneto-electric induction. In this he was eminently successful,
and we shall indicate the method by which the laws of induced
currents may be deduced from Weber's formula. But we must
observe,, that the circumstance that a law deduced from the pheno
mena discovered by Ampere is able also to account for the pheno
mena afterwards discovered by Faraday does not give so much
additional weight to the evidence for the physical truth of the law
as we might at first suppose.
For it has been shewn by Helmholtz and Thomson (see Art. 543),
that if the phenomena of Ampere are true, and if the principle of
the conservation of energy is admitted, then the phenomena of in
duction discovered by Faraday follow of necessity. Now Weber's
law, with the various assumptions about the nature of electric
currents which it involves, leads by mathematical transformations
to the formula of Ampere. Weber's law is also consistent with the
principle of the conservation of energy in so far that a potential
exists, and this is all that is required for the application of the
principle by Helmholtz and Thomson. Hence we may assert, even
before making any calculations on the subject, that Weber's law
will explain the induction of electric currents. The fact,, therefore,
that it is found by calculation to explain the induction of currents,
leaves the evidence for the physical truth of the law exactly where
it was.
On the other hand, the formula of Gauss, though it explains the
phenomena of the attraction of currents, is inconsistent with the
principle of the conservation of energy, and therefore we cannot
assert that it will explain all the phenomena of induction. In fact,
it fails to do so, as we shall see in Art. 859.
857.] We must now consider the electromotive force tending to
produce a current in the element els', due to the current in ds, when
ds is in motion, and when the current in it is variable.
According to Weber, the action on the material of the conductor
of which ds' is an element, is the sum of all the actions on the
electricity which it carries. The electromotive force, on the other
hand, on the electricity in dts't is the difference of the electric forces
acting on the positive and the negative electricity within it. Since
all these forces act in the line joining the elements, the electro
motive force on ds' is also in this line, and in order to obtain the
electromotive force in the direction of ds' we must resolve the force
in that direction. To apply Weber's formula, we must calculate
the various terms which occur in it, on the supposition that the
858.] WEBER'S THEORY OF INDUCED CURRENTS. 433
element ds is in motion relatively to els', and that the currents in
both elements vary with the time. The expressions thus found
will contain terms involving* v2, vv' ', v'2, v, ?/, and terms not involv
ing v or v', all of which are multiplied by ee'. Examining, as we
did before, the four values of each term, and considering first the
mechanical force which arises from the sum of the four values, we
find that the only term which we must take into account is that
involving the product vv' ' ee' '.
If we then consider the force tending to produce a current in the
second element, arising from the difference of the action of the first
element on the positive and the negative electricity of the second
element, we find that the only term which we have to examine is
that which involves vee'. We may write the four terms included in
2 (veef), thus
e' (ve -f vl tfj) and e\ (ve + vl e^.
Since e'-\-e\ = 0, the mechanical force arising from these terms is
zero, but the electromotive force acting on the positive electricity e'
is (ve + v-± e^, and that acting on the negative electricity e\ is equal
and opposite to this.
858.] Let us now suppose that the first element ds is moving
relatively to ds' with velocity V in a certain direction, and let us
A A
denote by Yds and Yds' ', the angle between the direction of V and
that of ds and of ds' respectively, then the square of the relative
velocity, u9 of two electric particles is
u2 = v2+v'2+72-2vv'cose+27vcosFds-27v'cos7cti. (25)
The term in vv' is the same as in equation (3). That in v, on which
the electromotive force depends, is
A
2 Fv cos Yds.
We have also for the value of the time- variation of r in this case
c) r dr fdr dr
— = v --- + >o'— + —, (26)
^t ds ds dt
where ^- refers to the motion of the electric particles, and ^- to
that of the material conductor. If we form the square of this quan
tity, the term involving vif, on which the mechanical force depends,
is the same as before, in equation (5), and that involving v, on which
the electromotive force depends, is
dr dr
2v-r-rr>
ds dt
VOL. ii. r f
434 ACTION AT A DISTANCE. [859.
Differentiating (26) with respect to t, we find
dv dr , dv' dr d2r
*v~foTs + v ^di^di2'
We find that the term involving vv' is the same as before in (6).
The term whose sign alters with that of v is -=7- -=- •
dt ds
859.] If we now calculate by the formula of Gauss (equation (18)),
the resultant electrical force in the direction of the second element
ds' y arising from the action of the first element ds, we obtain
1 A A A A
-y dsds'i V (2 cos Yds — 3 cos Vr cos r ds) coerdi. (28)
As in this expression there is no term involving the rate of va
riation of the current i, and since we know that the variation of
the primary current produces an inductive action on the secondary
circuit, we cannot accept the formula of Gauss as a true expression
of the action between electric particles.
860.] If, however, we employ the formula of Weber, (19), we
obtain \ drdi .drdr.dr f .
(29)
., — ,
r2 S ds dt ds dt> ds
dr dr d ,i\ 7 7 , ,QA>.
or -Y -j-, -j- (-) dsds'. (30)
ds ds dt\r'
If we integrate this expression with respect to s and /, we obtain
for the electromotive force on the second circuit
d . CCl dr dr , .
•s'JJ ;***?•
Now, when the first circuit is closed,
d2r
ds ds'
= 0.
/*! dr dr , f A dr dr d2r \ , /*cose T
Hence / - T -^ ds = / (- — — + ~-7-7) ds = - I - - ds. (32)
J r ds ds' J V ds ds dsds'' J r
But fj^^dsds/= M, by Arts. 423, 524. (33)
Hence we may write the electromotive force on the second circuit
-*<•'*>• (34)
which agrees with what we have already established by experiment ;
Art. 539.
863.] KEYSTONE OF ELECTRODYNAMICS. 435
On Weber s Formula^ considered as resulting from an Action transmitted
from one Electric Particle to the other with a Constant Velocity.
861.] In a very interesting letter of Gauss to W. Weber * he
refers to the electrodynamic speculations with which he had been
occupied long before, and which he would have published if he could
then have established that which he considered the real keystone
of electrodynamics, namely, the deduction of the force acting be
tween electric particles in motion from the consideration of an action
between them, not instantaneous, but propagated in time, in a
similar manner to that of light. He had not succeeded in making
this deduction when he gave up his electrodynamic researches, and
he had a subjective conviction that it would be necessary in the
first place to form a consistent representation of the manner in
which the propagation takes place.
Three eminent mathematicians have endeavoured to supply this
keystone of electrodynamics.
862. J In a memoir presented to the Royal Society of Gottingen
in 1858, but afterwards withdrawn, and only published in Poggen-
dorff's Annalen in 1867, after the death of the author, Bernhard
Riemann deduces the phenomena of the induction of electric cur
rents from a modified form of Poisson's equation
where Fis the electrostatic potential, and a a velocity.
This equation is of the same form as those which express the
propagation of waves and other disturbances in elastic media. The
author, however, seems to avoid making explicit mention of any
medium through which the propagation takes place.
The mathematical investigation given by Riemann has been ex
amined by Clausiusf, who does not admit the soundness of the
mathematical processes, and shews that the hypothesis that potential
is propagated like light does not lead either to the formula of Weber,
or to the known laws of electrodynamics.
863.] Clausius has also examined a far more elaborate investiga
tion by C. Neumann on the ' Principles of Electrodynamics' J. Neu
mann, however, lias pointed out§ that his theory of the transmission
of potential from one electric particle to another is quite different
from that proposed by Gauss, adopted by Riemann, and criticized
* March 19, 1845, WerJse, bd. v. 629. £ Tubingen, 1868.
t Pogg., bd. cxxxv. 612. § Mathematische Annalen, i. 317.
436 ACTION AT A DISTANCE. [864.
by Clausius, in which the propagation is like that of light. There
is, on the contrary, the greatest possible difference between the
transmission of potential, according to Neumann, and the propaga
tion of light.
A luminous body sends forth light in all directions, the intensity
of which depends on the luminous body alone, and not on the
presence of the body which is enlightened by it.
An electric particle, on the other hand, sends forth a potential,
ed
the value of which, — , depends not only on <?, the emitting particle,
but on e' , the receiving particle, and on the distance r between the
particles at the instant of emission.
In the case of light the intensity diminishes as the light is pro
pagated further from the luminous body ; the emitted potential
flows to the body on which it acts without the slightest alteration
of its original value.
The light received by the illuminated body is in general only a
fraction of that which falls on it ; the potential as received by the
attracted body is identical with, or equal to, the potential which
arrives at it.
Besides this, the velocity of transmission of the potential is not,
like that of light, constant relative to the aether or to space, but
rather like that of a projectile, constant relative to the velocity of
the emitting particle at the instant of emission.
It appears, therefore, that in order to understand the theory of
Neumann, we must form a very different representation of the pro
cess of the transmission of potential from that to which we have
been accustomed in considering the propagation of light. Whether
it can ever be accepted as the ' construirbar Vorstellung' of the
process of transmission, which appeared necessary to Gauss, I cannot
say, but I have not myself been able to construct a consistent
mental representation of Neumann's theory.
864.] Professor Betti*, of Pisa, has treated the subject in a
different way. He supposes the closed circuits in which the electric
currents flow to consist of elements each of which is polarized
periodically, that is, at equidistant intervals of time. These polar
ized elements act on one another as if they were little magnets
whose axes are in the direction of the tangent to the circuits. The
periodic time of this polarization is the same in all electric cir
cuits. Betti supposes the action of one polarized element on an-
* Nuovo Cimento, xxvii (1868).
866.] A MEDIUM NECESSARY. 437
other at a distance to take place, not instantaneously, but after a
time proportional to the distance between the elements. In this
way he obtains expressions for the action of one electric circuit on
another, which coincide with those which are known to be true.
Clausius, however, has, in this case also, criticized some parts of
the mathematical calculations into which we shall not here enter.
865.] There appears to be, in the minds of these eminent men,
some prejudice, or a priori objection, against the hypothesis of a
medium in which the phenomena of radiation of light and heat,
and the electric actions at a distance take place. It is true that at
one time those who speculated as to the causes of physical pheno
mena, were in the habit of accounting for each kind of action at a
distance by means of a special sethereal fluid, whose function and
property it was to produce these actions. They filled all space
three and four times over with aethers of different kinds, the pro
perties of which were invented merely to ' save appearances,' so that
more rational enquirers were willing rather to accept not only New
ton's definite law of attraction at a distance, but even the dogma of
Cotes "*, that action at a distance is one of the primary properties of
matter, and that no explanation can be more intelligible than this
fact. Hence the undulatory theory of light has met with much
opposition, directed not against its failure to explain the pheno
mena, but against its assumption of the existence of a medium in
which light is propagated.
866.] We have seen that the mathematical expressions for electro-
dynamic action led, in the mind of Gauss, to the conviction that a
theory of the propagation of electric action in time would be found
to be the very key-stone of electrodynamics. Now we are unable
to conceive of propagation in time, except either as the flight of a
material substance through space, or as the propagation of a con
dition of motion or stress in a medium already existing in space.
In the theory of Neumann, the mathematical conception called
Potential, which we are unable to conceive as a material substance,
is supposed to be projected from one particle to another, in a manner
which is quite independent of a medium, and which, as Neumann
has himself pointed out, is extremely different from that of the pro
pagation of light. In the theories of Riemann and Betti it would
appear that the action is supposed to be propagated in a manner
somewhat more similar to that of light.
But in all of these theories the question naturally occurs : — If
* Preface to Newton's Principia, 2nd edition.
438 ACTION AT A DISTANCE. [866.
something is transmitted from one particle to another at a distance,
what is its condition after it has left the one particle and before
it has reached the other ? If this something is the potential energy
of the two particles, as in Neumann's theory, how are we to con
ceive this energy as existing in a point of space, coinciding neither
with the one particle nor with the other ? In fact, whenever energy
is transmitted from one body to another in time, there must be
a medium or substance in which the energy exists after it leaves
one body and before it reaches the other, for energy, as Torricelli *
remarked, ' is a quintessence of so subtile a nature that it cannot be
contained in any vessel except the inmost substance of material
things.' Hence all these theories lead to the conception of a medium
in which the propagation takes place, and if we admit this medium
as an hypothesis, I think it ought to occupy a prominent place in
our investigations, and that we ought to endeavour to construct a
mental representation of all the details of its action, and this has
been my constant aim in this treatise.
* Lezioni Accademiche (Firenze, 1715), p. 25.
INDEX.
The References are to the Articles.
ABERRATION of light, 78.
Absorption, electric, 53, 227, 329.
— of light, 798.
Accumulators or condensers, 50, 226-228.
Action at a distance, 105, 641-646, 846-
866.
Acyclic region, 19, 113.
^Ether, 782 n.
Airy, Sir G. B., 454, 830.
Ampere, Andr£ Marie, 482, 502-528,
638, 687, 833, 846.
Anion, 237.
Anode, 237.
Arago's disk, 668, 669.
Astatic balance, 504.
Atmospheric electricity, 221.
Attraction, electric, 27, 38, 103.
— explained by stress in a medium, 105.
Barclay and Gibson, 229, 789.
Battery, voltaic, 232.
Beetz, W., 255, 265, 442.
Betti, E., 173, 864.
Bifilar suspension, 459.
Bismuth, 425.
Borda, J. C., 3.
Bowl, spherical, 176-181.
Bridge, Wheatstone's*. 347, 756, 775, 778.
— electrostatic, 353.
Bright, Sir C., and Clark, 354, 367.
Brodie, Sir B. C., 359.
Broun, John Allan, 462.
Brush, 56.
Buff, Heinrich. 271, 368.
Capacity (electrostatic), 50, 226.
— of a condenser, 50, 87, 102, 196, 227-
229, 771, 774-780.
Capacity, calculation of, 102, 196.
— measurement of, 227-229.
— in electromagnetic measure, 774,
775.
Capacity (electromagnetic) of a coil, 706,
756, 778, 779.
Cathode, 237.
Cation, 237.
Cauchy, A. L., 827.
Cavendish, Henry, 38.
Cayley, A., 553.
Centrobaric, 101.
Circuits, electric, 578-584.
Circular currents, 694-706.
— solid angle subtended by, 695.
Charge, electric, 31.
Clark, Latimer, 358, 629, 725.
Classification of electrical quantities, 620-
629.
Clausius, R., 70, 256, 863.
Clifford, W. K., 138.
Coefficients of electrostatic capacity and
induction, 87, 102.
— of potential, 87.
— of resistance and conductivity, 297,
298.
— of induced magnetization, 426.
— of electromagnetic induction, 755.
— of self-induction, 756, 757.
Coercive force, 424, 444.
Coils, resistance, 335-344.
— electromagnetic, 694-706.
— measurement of, 708.
— comparison of, 752-757.
Comparison of capacities, 229.
— of coils, 752-757.
— of electromotive forces, 358.
— of resistances, 345-358.
Concentration, 26, 77.
Condenser, 50, 226-228.
* Sir Charles Wheatstone, in his paper on ' New Instruments and Processes,' Phil.
Trans., 1843, brought this arrangement into public notice, with due acknowledgment
of the original inventor, Mr. S. Hunter Christie, who had described it in his paper on
'Induced Currents,' Phil. Trans., 1833, under the name of a Differential Arrange
ment. See the remarks of Mr. Latimer Clark in the Society of Telegraph Engineers,
May 8, 1872.
440
I N D E X.
Condenser, capacity of, 50, 87, 102, 196,
227-229, 771, 774-780.
Conduction, 29, 241-254.
Conduction, linear, 273-284.
— superficial, 294.
— in solids, 285-334.
— electrolytic, 255-265.
— in dielectrics, 325-334.
Conductivity, equations of, 298, 609.
— and opacity, 798.
Conductor, 29, 80, 86.
Conductors, systems of electrified, 84-94.
Confocal quadric surfaces, 147-154, 192.
Conjugate circuits, 538, 759.
— conductors, 282, 347.
— functions, 182-206.
— harmonics, 138.
Constants, principal, of a coil, 700, 753,
754.
Conservation of energy, 92, 242, 262, 543,
Contact force, 246.
Continuity in time and space, 7.
— equation of, 36, 295.
Convection, 55, 238, 248.
Convergence, 25.
Copper, 51, 360, 362, 761.
Cotes, Roger, 865.
Coulomb, C. A., 38, 74, 215, 223, 373.
Coulomb's law, 79, 80.
Crystal, conduction in, 297.
— magnetic properties of, 435, 436, 438.
— propagation of light in a, 794—797.
Gumming, James, 252.
Curl, 25.
Current, electric, 230.
— be.st method of applying, 744.
— induced, 582.
— steady, 232.
— thermoelectric, 249-254.
— transient, 232, 530, 536, 537, 582,
748, 758, 760, 771, 776.
Current- weigher, 726.
Cyclic region, 18, 113, 481.
Cylinder, electrification of, 189.
— magnetization of, 436, 438, 439.
— currents in, 682-690.
Cylindric coils, 676-681.
Damped vibrations, 732-742, 762.
Damper, 730.
Daniell's cell, .232, 272.
Dead beat galvanometer, 741.
Decrement, logarithmic, 736.
Deflexion, 453, 743.
Delambre, J. B. J., 3.
Dellmann, F., 221.
Density, electric, 64.
— of a current, 285.
— measurement of, 223.
Diamagnetism, 429, 440, 838.
Dielectric, 52, 109, 111, 229, 325-334,
366-370, 784.
[ Diffusion of magnetic force, 801.
Dip, 461.
Dipolar, 173, 381.
Dimensions, 2, 42, 87, 278, 620-629.
Directed quantities (or vectors), 10.
Directrix, 517.
Discharge, 55.
Discontinuity, 8.
Disk, 177.
— Arago's, 668, 669.
Displacement, electric, 60, 75, 76, 111,
328-334, 608, 783, 791.
Dygogram, 441.
Earnshaw, S., 116.
Earth, magnetism of, 465-474.
Electric brush, 56.
— charge, 31.
— conduction, 29.
— convection, 211, 238, 248, 255, 259.
— current, 230.
— discharge, 55-57.
— displacement, 60, 75, 76, 111, 328-
324, 608, 783, 791.
— energy, 85.
— : glow, 55.
— induction, 28.
— machine, 207.
— potential, 70.
— spark, 57.
— tension, 48, 59, 107, 108, 111.
— wind, 55.
Electrode, 237.
Electrodynamic system of measurement,
526.
Electrodynamometer, 725.
Electrolysis, 236, 255-272.
Electrolyte, 237, 255.
Electrolytic conduction, 255-272, 363,
799.
— polarization, 257, 264-272.
Electromagnetic force, 475, 580, 583.
— measurement, 495.
— momentum, 585.
— observations, 730-780.
— and electrostatic units compared, 768—
780.
— rotation, 491.
Electromagnetism, dynamical theory of,
568-577.
Electrometers, 214-220.
Electromotive force, 49, 69, 111, 241,
246-254, 358, 569, 579.
Electrophorus, 208.
Electroscope, 33, 214.
Electrostatic measurements, 214-229.
— polarization, 59, 111.
— attraction, 103-111.
— system of units, 620, &c.
Electrotonic state, 540.
Elongation, 734.
Ellipsoid, 150, 302, 437, 439.
Elliptic integrals, 149, 437, 701.
Energy, 6, 85, 630-638, 782, 792.
I N D E X.
441
Equations of conductivity, 298, 609.
— of continuity, 35.
— of electric currents, 607.
— of total currents, 610.
— of electromagnetic force, 603.
— of electromotive force, 598.
— of Laplace, 77.
— of magnetization, 400, 605.
— of magnetic induction, 591.
— of Poisson, 77.
— of resistance, 297.
Equilibrium, points of, 112-117.
False magnetic poles, 468.
Faraday, M., his discoveries, 52, 55, 236,
255, 530, 531, 534, 546, 668, 806.
— his experiments, 28, 429, 530, 668.
— his methods, 37, 82, 122, 493, 528,
529, 541, 592, 594, 604.
— his speculations, 54, 60, 83, 107, 109,
245, 429, 502, 540, 547, 569, 645, 782.
Farad, 629.
Fechner, G. T., 231, 274, 848.
Felici, R., 536-539, 669.
Ferromagnetic, 425, 429, 844.
Field, electric, 44.
— electromagnetic, 585-619.
— of uniform force, 672.
First swing, 745.
Fizeau, H. L., 787.
Fluid, electric, 36, 37.
— incompressible, 61, 111, 295, 329, 334.
— magnetic, 380.
Flux, 12.
Force, electromagnetic, 475, 580, 583.
— electromotive, 49, 69, 111, 233, 241,
246-254, 358, 569, 579, 595, 598.
— mechanical, 92, 93, 103-111, 174, 580,
602.
— measurement of, 6.
— acting at a distance, 105.
— lines of, 82, 117-123, 404.
Foucault, L., 787.
Fourier, J. B. J., 2w, 243, 332, 333, 801-
805.
Galvanometer, 240, 707.
— differential, 346.
— sensitive, 717.
— standard, 708.
— observation of, 742-751.
Gases, electric discharge in, 55-77, 370.
— resistance of, 369.
Gassiot, J. P., 57.
Gaugain, J. M., 366, 712.
Gauge electrometer, 218.
Gauss, C. F., 18, 70, 131, 140, 144, 409,
421, 454, 459, 470, 706, 733, 744, 851.
Geometric mean distance, 691-693.
Geometry of position, 421.
Gibson and Barclay, 229, 789.
Gladstone, Dr. J. H., 789.
Glass, 51, 271, 368.
Glow, electric, 55.
Grassmann, H., 526, 687.
Grating, electric effect of, 203.
Green, George, 70, 89, 318, 439.
Green's function, 88, 101.
— theorem, 100.
Groove, electric effect of, 199.
Grove, Sir W. R., 272, 779.
Guard-ring, 201, 217, 228.
Gutta-percha, 51, 367.
Hamilton, Sir W. Rowan, 10, 561.
Hard iron, 424, 444.
Harris, Sir W. Snow, 38, 216.
Heat, conduction of, 801.
— generated by the current, 242, 283,
299.
— specific, of electricity, 253.
Helix, 813.
Helmholtz, H., 88, 100, 202, 421, 543,
713, 823, 854.
Heterostatic electrometers, 218.
Hockin, Charles, 352, 360, 800.
Holtz, W., electrical machine, 212.
Hornstein, Karl, 471 n.
Huygens, Christian, 782.
Hydraulic ram, 550.
Hyposine, 151.
Idiostatic electrometers, 218.
Images, electric, 119, 155-181, 189.
— magnetic, 318.
— moving, 662.
Imaginary magnetic matter, 380.
Induced currents, 528-552.
— in a plane sheet, 656-669.
— Weber's theory of, 856.
Induced magnetization, 424-448.
Induction, electrostatic, 28, 75, 76, 111.
— magnetic, 400.
Inertia, electric, 550.
— moments and products of, 565.
Insulators, 29.
Inversion, electric, 162-181, 188, 316.
Ion, 237, 255.
Iron, 424.
— perchloride of, 809.
Irreconcileable curves, 20, 421.
Jacobi, M. H., 336.
Jenkins, William, 546. See Phil Mag.,
1834, pt. ii, p. 351.
Jenkin, Fleeming, 763, 774.
Jochmann, E., 669.
Joule, J. P., 242, 262, 448, 457, 463, 726,
767.
'Keystone of electrodynamics,' 861.
Kinetics, 553-565.
Kirchhoff, Gustav, 282, 316, 439, 758.
Kohlrausch, Rudolph, 265, 365, 723, 771.
442
INDEX.
Lagrange's (J. L.) dynamical equations,
553-565.
Lame", G., 17, 147.
Lamellar magnet, 412.
Laplace, P. S., 70.
Laplace's coefficients, 128-146.
— equation, 26, 77, 144, 301.
— expansion, 140.
Leibnitz, G. W., 18, 424.
Lenz, E., 265, 530, 542.
Light, electromagnetic theory of, 781-805.
— and magnetism, 806-831.
Line-density, 64, 81.
integral, 16-20.
— of electric force, 69, 622.
— of magnetic force, 401, 481, 498, 499,
590, 606, 607, 622.
Lines of equilibrium, 112.
— of flow, 22, 293.
— of electric induction, 82, 117-123.
— of magnetic induction, 404, 489, 529,
541, 597, 702.
Linnaeus, C., 23.
Liouville, J., 173, 176.
Listing, J. B., 18, 23, 421.
Lorenz, L., 805 n.
Loschmidt, J., 5.
Magnecrystallic phenomena, 425, 435,
839.
Magnet, its properties, 371.
— direction of axis, 372-390.
— magnetic moment of, 384, 390.
— centre and principal axes, 392.
— potential energy of, 389.
Magnetic action of light, 806.
— disturbances, 473.
— force, law of, 374.
direction of, 372, 452.
intensity of, 453.
— induction, 400.
Magnetic ' matter,' 380.
— measurements, 449-464.
— poles, 468.
— survey, 466.
— variations, 472.
Magnetism of ships, 441.
— terrestrial, 465-474.
Magnetization, components of, 384.
— induced, 424-430.
— Ampere's theory of, 638, 833.
— Poisson's theory of, 429.
_ Weber's theory of, 442, 838.
Magnus' (G.) law, 251.
Mance's, Henry > method, 357.
Matthiessen, Aug., 352, 360.
Measurement, theory of, 1.
— of result of electric force, 38.
— of electrostatic capacity, 226-229.
— of electromotive force or potential,
216, 358.
— of resistance, 335-357.
— of constant currents, 746.
— of transient currents, 748.
Measurement of coils, 70S, 752-757.
— magnetic, 449-464.
Medium, electromagnetic, 866.
— lummiferous, 806.
Mercury, resistance of, 361.
Metals, resistance of, 363.
Michell, John, 38.
Miller, W. H., 23.
Mirror method, 450.
Molecular charge of electricity, 259.
— currents, 833.
— standards, 5.
— vortices, 822.
Molecules, size of, 5.
— electric, 260.
— magnetic, 430, 832-845.
Moment, magnetic, 384.
— of inertia, 565.
Momentum, 6.
— electrokinetic, 578, 585.
Mossotti, O. F., 62.
Motion, equations of, 553-565.
Moving axes, 600.
— conductors, 602.
— images, 662.
Multiple conductors, 276, 344.
— functions. 9.
Multiplication, method of, 747, 751.
Neumann, F. E., coefficient of magnetiza
tion, 430.
— magnetization of ellipsoid, 439.
— theory of induced currents, 542.
Neumann, C. G., 190, 830, 863.
Nicholson's Eevolving Doubler, 209.
Nickel, 425.
Null methods, 214, 346, 503.
Orsted, H. C., 239, 475.
Ohm, G. S., 241, 333.
Ohm's Law, 241.
Ohm, the, 338, 340, 629.
Opacity, 798,
Ovary ellipsoid, 152.
Paalzow, A., 364.
Paraboloids, confocal, 154.
Paramagnetic (same as Ferromagnetic),
425, 429, 844.
Peltier, A., 249.
Periodic functions, 9.
Periphractic region, 22, 113.
Permeability, magnetic, 428, 614.
Phillips, S. E., 342.
Plan of this Treatise, 59.
Plane current-sheet, 656-669.
Planetary ellipsoid, 151.
Platymeter, electro-, 229.
Plucker, Julius, 839.
Points of equilibrium, 112.
Poisson, S. D , 155, 431, 437, 674.
Poisson's equation, 77, 148.
INDEX.
443
Poisson's theory of magnetism, 427, 429,
431, 441, 832.
— theory of wave-propagation, 784.
Polar definition of magnetic force, 398.
Polarity, 381.
Polarization, electrostatic, 59, 111.
— electrolytic, 257, 264-272.
— magnetic, 381.
— of light, 381, 791.
— circular, 813,
Poles of a magnet, 373.
magnetic of the earth, 468.
Positive and negative, conventions about,
23, 27, 36, 37, 63, 68-81, 231, 374, 394,
417, 489, 498.
Potential, 16.
— electric, 45, 70, 220.
— magnetic, 383, 391.
— of magnetization, 412.
— of two circuits, 423.
— of two circles, 698.
Potential, vector-, 405, 422, 590, 617,
657.
Principal axes, 299, 302.
Problems, electrostatic, 155-205.
— electrokinematic, 306-333.
— magnetic, 431-441.
— electromagnetic, 647-706.
Proof of the law of the Inverse Square,
74.
Proof plane, 223.
Quadrant electrometer, 219.
Quadric surfaces, 147-154.
Quantity, expression for a physical, 1.
Quantities, classification of electromag
netic, 620-629.
Quaternions, 11, 303, 490, 522, 618.
Quinke, G., 316 n.
Radiation, forces concerned in, 792.
Rankine, W. J. M., 115, 831.
Ray of electromagnetic disturbance, 791.
Reciprocal properties, electrostatic, 88.
— electrokinematic, 281, 348.
— magnetic, 421, 423.
— electromagnetic, 536,
_ kinetic, 565.
Recoil, method of, 750.
Residual charge, 327-334.
— magnetization, 444.
Replenisher, 210.
Resistance of conductors, 51, 275.
— tables of, 362-365.
— equations of, 297.
— unit of, 758-767.
— electrostatic measure of, 355, 780.
Resultant electric force at a point, 68.
Riemann, Bernhard, 421, 862.
Right and left-handed systems of axes,
23,498, 501.
— crcularly-polarized rays, 813.
Ritchie, W., 542.
Ritter's (J. W.) Secondary Pile, 271.
Rotation of plane of polarization, 806.
— magnetism, a phenomenon of, 821.
Riihlmann, R.., 370.
Rule of electromagnetic direction, 477,
494, 496.
Scalar, 11.
Scale for mirror observations, 450.
Sectorial harmonic, 132, 138.
Seebeck, T. J., 250.
Selenium, 51, 362.
Self-induction, 7.
— measurement of, 756, 778, 779.
— coil of maximum, 706.
Sensitive galvanometer, 717.
Series of observations, 746, 750.
Shell, magnetic, 409, 484, 485, 606, 652,
670, 694, 696.
Siemens, C. W., 336, 361.
Sines, method of, 455, 710.
Singular points, 128.
Slope, 17.
Smee, A., 272.
Smith, Archibald, 441.
Smith, W. R., 123, 316.
Soap bubble, 125.
Solenoid, magnetic, 407.
— electromagnetic, 676-681, 727.
Solenoidal distribution, 21, 82, 407.
Solid angle, 409, 417-422, 485, 695.
Space- variation, 17, 71, 835.
Spark, 57, 370.
Specific inductive capacity, 52, 83, 94,
111, 229, 325, 334, 627, 788.
— conductivity, 278, 627.
— resistance, 277, 627.
— heat of electricity, 253.
Sphere, 125.
Spherical harmonics, 128-146, 391, 431.
Spiral, logarithmic, 731.
Standard electrometer, 217.
— galvanometer, 708.
Stokes, G. G., 24, 115, 784.
Stoney, G. J., 5.
Stratified conductors, 319.
Stress, electrostatic, 107, 111.
— electrokinetic, 641, 645, 646.
Strutt, Hon. J. W., 102, 306.
Surface-integral, 15, 21, 75, 402.
density, 64, 78, 223.
Surface, equipotential, 46.
— electrified, 78.
Suspended coil, 721-729.
Suspension, bifilar, 45S.
— Joule's, 463.
— Thomson's, 721.
— unifilar, 449.
Tables of coefficients of a coil, 700.
— of dimensions, 621-629.
— of electromotive force, 358.
— of magnetic rotation, 830.
444
I N D E X.
Tables for magnetization of a cylinder,
439.
— of resistance, 363-365.
— of velocity of light and of electromag
netic disturbance, 787-
— of temporary and residual magnetiza
tion, 445.
Tait, P. G., 25, 254, 387, 522, 687,
731.
Tangent galvanometer, 710.
Tangents, method of, 454, 710.
Telegraph cable, 332, 689.
Temporary magneti/ation, 444.
Tension, electrostatic, 48, 59, 107, 108.
— electromagnetic, 645, 646.
Terrestrial magnetism, 465-474.
Thalen, Tobias Robert, 430.
Theorem, Green's, 100.
— Earnshaw's, 116.
— Coulomb's, 80.
— Thomson's, 98.
— Gauss', 144, 409.
Theory of one fluid, 37.
— of two fluids, 36.
— of magnetic matter, 380.
— of magnetic molecules, 430, 832-845.
— of molecular currents, 833.
— of molecular vortices, 822.
— of action at a distance, 105, 641-646,
846-866.
Thermo-electric currents, 249-254.
Thickness of galvanometer wire, 716,
719.
Thomson, Sir William,
— electric images, 43, 121, 155-181,
173.
— experiments, 51, 57, 248, 369, 772.
— instruments, 127, 201, 210, 211, 216-
222, 272, 722, 724.
— magnetism, 318, 398, 400, 407-416,
428.
— resistance, 338, 351, 356, 763.
— thermo-electricity, 207, 242, 249, 252,
253.
— theorems, 98, 138, 263, 299, 304,
652.
— theory of electricity, 27, 37, 543, 831,
856.
— vortex motion, 20, 100, 487, 702.
Thomson and Tait's Natural Philoso
phy, 132, 141, 144, 162, 303, 553,
676.
Time, periodic of vibration, 456, 738.
Time-integral, 541, 558.
Torricelli, Evangelista, 866.
Torsion-balance, 38, 215, 373, 726.
Transient currents, 232, 530, 536,* 537,
582, 748, 758, 760, 771, 776.
Units of physical quantities, 2.
— three fundamental, 3.
— derived, 6.
— electrostatic, 41, 625.
— magnetic, 374, 625.
— electrodynamic, 526.
— electromagnetic, 526, 620.
— classification of, 620-629.
— practical, 629.
— of resistance, 758-767.
— ratios of the two systems, 768-780.
Variation of magnetic elements, 472.
Varley, C. F., 210, 271, 332, 369.
Vector, 1 0.
Vector-potential, 405, 422, 590, 617, 657.
Velocity represented by the unit of re
sistance, 338, 628, 758.
by the ratio of electric units, 768-
780.
— of electromagnetic disturbance, 784.
— of light, 787.
— of the electric current, 569.
Verdet, M. E., 809, 830.
Vibration, time of, 456, 738.
Volt, 629.
Volta, A., 246.
Voltameter, 237.
Vortices, molecular, 822-831.
Water, resistance of, 365.
Wave-propagation, 784, 785.
Weber, W., 231, 338, 346.
— electrodynamometer, 725.
— induced magnetism, 442-448, 838.
— unit of resistance, 760-762.
— ratio of electric units, 227, 771.
— electrodynamic formula, 846-861.
Wertheim, W., 447.
Wheatstone's Bridge, 347-
— electrostatic, 353, 756, 775, 778.
Whewell, W., 237.
Wiedemann, G., 236, 370, 446, 447.
Wind, electric, 55.
Wippe, 775.
Work, 6.
Zero reading, 735.
Zonal harmonic, 132.
FILA1TE g
VOL. II .
Sl&clricily, Vol. 2Z.
FIG. XTV.
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